An application of integral...
Transcript of An application of integral...
![Page 1: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/1.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 1 / 18
An application of integral quaternionsBolyai Institute — Szeged, April 1, 2009
Lee M. Goswick, Emil W. Kiss, Gábor Moussong, Nándor Simányi
[email protected] University of Birmingham, Alabama, [email protected] Eötvös University, Budapest, [email protected] Eötvös University, Budapest, [email protected] University of Birmingham, Alabama, USA
![Page 2: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/2.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n
![Page 3: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/3.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonal
![Page 4: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/4.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonaland have the same length.
![Page 5: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/5.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonaland have the same length.The number k is the dimension of the icube.
![Page 6: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/6.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonaland have the same length.The number k is the dimension of the icube.
The subgroup {∑
aivi : ai ∈ Z} is a cubic lattice in Zn.
![Page 7: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/7.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonaland have the same length.The number k is the dimension of the icube.
The subgroup {∑
aivi : ai ∈ Z} is a cubic lattice in Zn.
Terminology: norm is length squared, so it is an integer.
![Page 8: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/8.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonaland have the same length.The number k is the dimension of the icube.
The subgroup {∑
aivi : ai ∈ Z} is a cubic lattice in Zn.
Terminology: norm is length squared, so it is an integer.
Main questionsConstruction: describe all icubes with given k and n.
![Page 9: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/9.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonaland have the same length.The number k is the dimension of the icube.
The subgroup {∑
aivi : ai ∈ Z} is a cubic lattice in Zn.
Terminology: norm is length squared, so it is an integer.
Main questionsConstruction: describe all icubes with given k and n.Counting: how many are there with a given length?
![Page 10: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/10.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonaland have the same length.The number k is the dimension of the icube.
The subgroup {∑
aivi : ai ∈ Z} is a cubic lattice in Zn.
Terminology: norm is length squared, so it is an integer.
Main questionsConstruction: describe all icubes with given k and n.Counting: how many are there with a given length?Extension: which ones can be extended
![Page 11: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/11.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonaland have the same length.The number k is the dimension of the icube.
The subgroup {∑
aivi : ai ∈ Z} is a cubic lattice in Zn.
Terminology: norm is length squared, so it is an integer.
Main questionsConstruction: describe all icubes with given k and n.Counting: how many are there with a given length?Extension: which ones can be extended (by adding vectors,that is, increasing the dimension)?
![Page 12: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/12.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonaland have the same length.The number k is the dimension of the icube.
The subgroup {∑
aivi : ai ∈ Z} is a cubic lattice in Zn.
Terminology: norm is length squared, so it is an integer.
Main questionsConstruction: describe all icubes with given k and n.Counting: how many are there with a given length?Extension: which ones can be extended (by adding vectors,that is, increasing the dimension)?
In the present work, n = 3 and k = 1, 2.
![Page 13: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/13.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 2 / 18
Icubes
An icube (integral cube) is a sequence (v1, . . . , vk )of nonzero vectors in Z
n that are pairwise orthogonaland have the same length.The number k is the dimension of the icube.
The subgroup {∑
aivi : ai ∈ Z} is a cubic lattice in Zn.
Terminology: norm is length squared, so it is an integer.
Main questionsConstruction: describe all icubes with given k and n.Counting: how many are there with a given length?Extension: which ones can be extended (by adding vectors,that is, increasing the dimension)?
In the present work, n = 3 and k = 1, 2.The case n = 3 and k = 3 was known before.
![Page 14: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/14.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w
![Page 15: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/15.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w is an integer.
![Page 16: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/16.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w is an integer.
Proof
The volume of the cube spanned by u, v , w is d3,
![Page 17: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/17.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w is an integer.
Proof
The volume of the cube spanned by u, v , w is d3,but it is also det(u, v , w),
![Page 18: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/18.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w is an integer.
Proof
The volume of the cube spanned by u, v , w is d3,but it is also det(u, v , w), hence d3 is an integer.
![Page 19: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/19.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w is an integer.
Proof
The volume of the cube spanned by u, v , w is d3,but it is also det(u, v , w), hence d3 is an integer.But d2 is also an integer,
![Page 20: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/20.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w is an integer.
Proof
The volume of the cube spanned by u, v , w is d3,but it is also det(u, v , w), hence d3 is an integer.But d2 is also an integer, since u, v , w ∈ Z
3.
![Page 21: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/21.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w is an integer.
Proof
The volume of the cube spanned by u, v , w is d3,but it is also det(u, v , w), hence d3 is an integer.But d2 is also an integer, since u, v , w ∈ Z
3.Thus d = d3/d2 is rational,
![Page 22: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/22.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w is an integer.
Proof
The volume of the cube spanned by u, v , w is d3,but it is also det(u, v , w), hence d3 is an integer.But d2 is also an integer, since u, v , w ∈ Z
3.Thus d = d3/d2 is rational, hence it is an integer.
![Page 23: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/23.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w is an integer.
Proof
The volume of the cube spanned by u, v , w is d3,but it is also det(u, v , w), hence d3 is an integer.But d2 is also an integer, since u, v , w ∈ Z
3.Thus d = d3/d2 is rational, hence it is an integer.
DefinitionPrimitive icube: The nk components of v1, . . . , vk are coprime.
![Page 24: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/24.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 3 / 18
The case k = 3
Observation
If (u, v , w) is an icube in Z3, then the common length d
of u, v , w is an integer.
Proof
The volume of the cube spanned by u, v , w is d3,but it is also det(u, v , w), hence d3 is an integer.But d2 is also an integer, since u, v , w ∈ Z
3.Thus d = d3/d2 is rational, hence it is an integer.
DefinitionPrimitive icube: The nk components of v1, . . . , vk are coprime.
It is clearly sufficient to construct all primitive icubes.
![Page 25: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/25.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 4 / 18
The construction of icubes for k = 3
Observation (Euler)For every m, n, p, q ∈ Z,
![Page 26: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/26.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 4 / 18
The construction of icubes for k = 3
Observation (Euler)For every m, n, p, q ∈ Z, the columns of E(m, n, p, q) =
m2 + n2 − p2 − q2 −2mq + 2np 2mp + 2nq2mq + 2np m2 − n2 + p2 − q2 −2mn + 2pq−2mp + 2nq 2mn + 2pq m2 − n2 − p2 + q2
yield an icube
![Page 27: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/27.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 4 / 18
The construction of icubes for k = 3
Observation (Euler)For every m, n, p, q ∈ Z, the columns of E(m, n, p, q) =
m2 + n2 − p2 − q2 −2mq + 2np 2mp + 2nq2mq + 2np m2 − n2 + p2 − q2 −2mn + 2pq−2mp + 2nq 2mn + 2pq m2 − n2 − p2 + q2
yield an icube with edge-length d = m2 + n2 + p2 + q2.
![Page 28: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/28.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 4 / 18
The construction of icubes for k = 3
Observation (Euler)For every m, n, p, q ∈ Z, the columns of E(m, n, p, q) =
m2 + n2 − p2 − q2 −2mq + 2np 2mp + 2nq2mq + 2np m2 − n2 + p2 − q2 −2mn + 2pq−2mp + 2nq 2mn + 2pq m2 − n2 − p2 + q2
yield an icube with edge-length d = m2 + n2 + p2 + q2.
This is called an Euler-matrix.
![Page 29: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/29.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 4 / 18
The construction of icubes for k = 3
Observation (Euler)For every m, n, p, q ∈ Z, the columns of E(m, n, p, q) =
m2 + n2 − p2 − q2 −2mq + 2np 2mp + 2nq2mq + 2np m2 − n2 + p2 − q2 −2mn + 2pq−2mp + 2nq 2mn + 2pq m2 − n2 − p2 + q2
yield an icube with edge-length d = m2 + n2 + p2 + q2.
This is called an Euler-matrix.
Theorem (A. Sárközy, 1961)This icube is primitive iff (m, n, p, q) = 1
![Page 30: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/30.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 4 / 18
The construction of icubes for k = 3
Observation (Euler)For every m, n, p, q ∈ Z, the columns of E(m, n, p, q) =
m2 + n2 − p2 − q2 −2mq + 2np 2mp + 2nq2mq + 2np m2 − n2 + p2 − q2 −2mn + 2pq−2mp + 2nq 2mn + 2pq m2 − n2 − p2 + q2
yield an icube with edge-length d = m2 + n2 + p2 + q2.
This is called an Euler-matrix.
Theorem (A. Sárközy, 1961)This icube is primitive iff (m, n, p, q) = 1 and d is odd.
![Page 31: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/31.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 4 / 18
The construction of icubes for k = 3
Observation (Euler)For every m, n, p, q ∈ Z, the columns of E(m, n, p, q) =
m2 + n2 − p2 − q2 −2mq + 2np 2mp + 2nq2mq + 2np m2 − n2 + p2 − q2 −2mn + 2pq−2mp + 2nq 2mn + 2pq m2 − n2 − p2 + q2
yield an icube with edge-length d = m2 + n2 + p2 + q2.
This is called an Euler-matrix.
Theorem (A. Sárközy, 1961)This icube is primitive iff (m, n, p, q) = 1 and d is odd.Every primitive icube can be obtained from a suitableEuler-matrix
![Page 32: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/32.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 4 / 18
The construction of icubes for k = 3
Observation (Euler)For every m, n, p, q ∈ Z, the columns of E(m, n, p, q) =
m2 + n2 − p2 − q2 −2mq + 2np 2mp + 2nq2mq + 2np m2 − n2 + p2 − q2 −2mn + 2pq−2mp + 2nq 2mn + 2pq m2 − n2 − p2 + q2
yield an icube with edge-length d = m2 + n2 + p2 + q2.
This is called an Euler-matrix.
Theorem (A. Sárközy, 1961)This icube is primitive iff (m, n, p, q) = 1 and d is odd.Every primitive icube can be obtained from a suitableEuler-matrix by permuting columns,
![Page 33: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/33.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 4 / 18
The construction of icubes for k = 3
Observation (Euler)For every m, n, p, q ∈ Z, the columns of E(m, n, p, q) =
m2 + n2 − p2 − q2 −2mq + 2np 2mp + 2nq2mq + 2np m2 − n2 + p2 − q2 −2mn + 2pq−2mp + 2nq 2mn + 2pq m2 − n2 − p2 + q2
yield an icube with edge-length d = m2 + n2 + p2 + q2.
This is called an Euler-matrix.
Theorem (A. Sárközy, 1961)This icube is primitive iff (m, n, p, q) = 1 and d is odd.Every primitive icube can be obtained from a suitableEuler-matrix by permuting columns,and by changing the sign of the last column.
![Page 34: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/34.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 5 / 18
Counting icubes for k = 3
Corollary (A. Sárközy, 1961)The number of primitive icubes with edge-length d is
![Page 35: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/35.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 5 / 18
Counting icubes for k = 3
Corollary (A. Sárközy, 1961)The number of primitive icubes with edge-length d is
f (d) = 8d∏
p prime, p|d
(
1 +1p
)
if d is odd,
![Page 36: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/36.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 5 / 18
Counting icubes for k = 3
Corollary (A. Sárközy, 1961)The number of primitive icubes with edge-length d is
f (d) = 8d∏
p prime, p|d
(
1 +1p
)
if d is odd, and 0 if d is even.
![Page 37: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/37.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 5 / 18
Counting icubes for k = 3
Corollary (A. Sárközy, 1961)The number of primitive icubes with edge-length d is
f (d) = 8d∏
p prime, p|d
(
1 +1p
)
if d is odd, and 0 if d is even.The number of all icubes with edge-length d is
∑
k |d f (k).
![Page 38: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/38.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 5 / 18
Counting icubes for k = 3
Corollary (A. Sárközy, 1961)The number of primitive icubes with edge-length d is
f (d) = 8d∏
p prime, p|d
(
1 +1p
)
if d is odd, and 0 if d is even.The number of all icubes with edge-length d is
∑
k |d f (k).
The proof is an application of the following well-known result.
![Page 39: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/39.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 5 / 18
Counting icubes for k = 3
Corollary (A. Sárközy, 1961)The number of primitive icubes with edge-length d is
f (d) = 8d∏
p prime, p|d
(
1 +1p
)
if d is odd, and 0 if d is even.The number of all icubes with edge-length d is
∑
k |d f (k).
The proof is an application of the following well-known result.
Theorem (Jacobi)If d is odd, then the number of solutions of
m2 + n2 + p2 + q2 = d (m, n, p, q ∈ Z)
is
![Page 40: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/40.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 5 / 18
Counting icubes for k = 3
Corollary (A. Sárközy, 1961)The number of primitive icubes with edge-length d is
f (d) = 8d∏
p prime, p|d
(
1 +1p
)
if d is odd, and 0 if d is even.The number of all icubes with edge-length d is
∑
k |d f (k).
The proof is an application of the following well-known result.
Theorem (Jacobi)If d is odd, then the number of solutions of
m2 + n2 + p2 + q2 = d (m, n, p, q ∈ Z)
is 8σ(d)
![Page 41: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/41.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 5 / 18
Counting icubes for k = 3
Corollary (A. Sárközy, 1961)The number of primitive icubes with edge-length d is
f (d) = 8d∏
p prime, p|d
(
1 +1p
)
if d is odd, and 0 if d is even.The number of all icubes with edge-length d is
∑
k |d f (k).
The proof is an application of the following well-known result.
Theorem (Jacobi)If d is odd, then the number of solutions of
m2 + n2 + p2 + q2 = d (m, n, p, q ∈ Z)
is 8σ(d) (here σ(d) is the sum of positive divisors of d).
![Page 42: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/42.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
![Page 43: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/43.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.
![Page 44: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/44.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.
![Page 45: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/45.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.
![Page 46: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/46.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
![Page 47: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/47.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
Theorem (1915 by R. D. Carmichael, may be earlier)
![Page 48: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/48.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
Theorem (1915 by R. D. Carmichael, may be earlier)
If a2 + b2 + c2 = d2,
![Page 49: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/49.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
Theorem (1915 by R. D. Carmichael, may be earlier)
If a2 + b2 + c2 = d2, where (a, b, c) = 1
![Page 50: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/50.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
Theorem (1915 by R. D. Carmichael, may be earlier)
If a2 + b2 + c2 = d2, where (a, b, c) = 1 and a is odd,
![Page 51: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/51.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
Theorem (1915 by R. D. Carmichael, may be earlier)
If a2 + b2 + c2 = d2, where (a, b, c) = 1 and a is odd, then
for some integers m, n, p, q
![Page 52: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/52.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
Theorem (1915 by R. D. Carmichael, may be earlier)
If a2 + b2 + c2 = d2, where (a, b, c) = 1 and a is odd, then
a = m2 + n2 − p2 − q2,
for some integers m, n, p, q
![Page 53: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/53.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
Theorem (1915 by R. D. Carmichael, may be earlier)
If a2 + b2 + c2 = d2, where (a, b, c) = 1 and a is odd, then
a = m2 + n2 − p2 − q2,
b = 2mq + 2np,
for some integers m, n, p, q
![Page 54: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/54.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
Theorem (1915 by R. D. Carmichael, may be earlier)
If a2 + b2 + c2 = d2, where (a, b, c) = 1 and a is odd, then
a = m2 + n2 − p2 − q2,
b = 2mq + 2np,
c = −2mp + 2nq,
for some integers m, n, p, q
![Page 55: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/55.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
Theorem (1915 by R. D. Carmichael, may be earlier)
If a2 + b2 + c2 = d2, where (a, b, c) = 1 and a is odd, then
a = m2 + n2 − p2 − q2,
b = 2mq + 2np,
c = −2mp + 2nq,
d = m2 + n2 + p2 + q2
for some integers m, n, p, q
![Page 56: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/56.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 6 / 18
Extension from k = 1 to k = 3
Which integral vectors can be put into an icube?
Necessary: The length must be an integer.Sufficient to deal with primitive vectors.Answer: All such vectors.We use the description of Pythagorean quadruples.
Theorem (1915 by R. D. Carmichael, may be earlier)
If a2 + b2 + c2 = d2, where (a, b, c) = 1 and a is odd, then
a = m2 + n2 − p2 − q2,
b = 2mq + 2np,
c = −2mp + 2nq,
d = m2 + n2 + p2 + q2
for some integers m, n, p, q (the first column of an Euler-matrix).
![Page 57: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/57.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
![Page 58: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/58.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal
![Page 59: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/59.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
![Page 60: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/60.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?
![Page 61: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/61.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.
![Page 62: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/62.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.Answer: All such pairs.
![Page 63: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/63.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.Answer: All such pairs. Indeed:
Elementary calculationIf u and v have length d ,
![Page 64: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/64.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.Answer: All such pairs. Indeed:
Elementary calculationIf u and v have length d , then w = (u × v)/d(cross product)
![Page 65: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/65.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.Answer: All such pairs. Indeed:
Elementary calculationIf u and v have length d , then w = (u × v)/d(cross product) is also an integral vector.
![Page 66: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/66.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.Answer: All such pairs. Indeed:
Elementary calculationIf u and v have length d , then w = (u × v)/d(cross product) is also an integral vector. Idea:
If x21 + x2
2 + x23 = d2 = y2
1 + y22 + y2
3
![Page 67: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/67.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.Answer: All such pairs. Indeed:
Elementary calculationIf u and v have length d , then w = (u × v)/d(cross product) is also an integral vector. Idea:
If x21 + x2
2 + x23 = d2 = y2
1 + y22 + y2
3 and x1y1 + x2y2 + x3y3 = 0,
![Page 68: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/68.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.Answer: All such pairs. Indeed:
Elementary calculationIf u and v have length d , then w = (u × v)/d(cross product) is also an integral vector. Idea:
If x21 + x2
2 + x23 = d2 = y2
1 + y22 + y2
3 and x1y1 + x2y2 + x3y3 = 0,then x2
3 y23 = x2
1 y21 + x2
2 y22 + 2x1x2y1y2,
![Page 69: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/69.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.Answer: All such pairs. Indeed:
Elementary calculationIf u and v have length d , then w = (u × v)/d(cross product) is also an integral vector. Idea:
If x21 + x2
2 + x23 = d2 = y2
1 + y22 + y2
3 and x1y1 + x2y2 + x3y3 = 0,then x2
3 y23 = x2
1 y21 + x2
2 y22 + 2x1x2y1y2, so
(x1y2 − x2y1)2
![Page 70: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/70.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.Answer: All such pairs. Indeed:
Elementary calculationIf u and v have length d , then w = (u × v)/d(cross product) is also an integral vector. Idea:
If x21 + x2
2 + x23 = d2 = y2
1 + y22 + y2
3 and x1y1 + x2y2 + x3y3 = 0,then x2
3 y23 = x2
1 y21 + x2
2 y22 + 2x1x2y1y2, so
(x1y2 − x2y1)2 = (x2
1 + x22 )(y2
1 + y22 ) − x2
3 y23
![Page 71: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/71.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 7 / 18
Extension from k = 2 to k = 3
Short name for k = 2 and n = 3
A twin pair is an ordered pair of vectors in Z3
that are orthogonal and have the same length.
Extension: Which twin pairs can be put into an icube?Necessary: The length must be an integer.Answer: All such pairs. Indeed:
Elementary calculationIf u and v have length d , then w = (u × v)/d(cross product) is also an integral vector. Idea:
If x21 + x2
2 + x23 = d2 = y2
1 + y22 + y2
3 and x1y1 + x2y2 + x3y3 = 0,then x2
3 y23 = x2
1 y21 + x2
2 y22 + 2x1x2y1y2, so
(x1y2 − x2y1)2 = (x2
1 + x22 )(y2
1 + y22 ) − x2
3 y23 is divisible by d2.
![Page 72: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/72.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
![Page 73: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/73.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
![Page 74: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/74.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
Then there exists an icube (u, v , w) with edge length m
![Page 75: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/75.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
Then there exists an icube (u, v , w) with edge length msuch that x = au + bv + cw for some a, b, c ∈ Z.
![Page 76: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/76.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
Then there exists an icube (u, v , w) with edge length msuch that x = au + bv + cw for some a, b, c ∈ Z.Thus the relative norm of x in this lattice is square-free.
![Page 77: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/77.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
Then there exists an icube (u, v , w) with edge length msuch that x = au + bv + cw for some a, b, c ∈ Z.Thus the relative norm of x in this lattice is square-free.
Theorem (GKMS)If x is primitive, then this cubic lattice is unique.
![Page 78: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/78.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
Then there exists an icube (u, v , w) with edge length msuch that x = au + bv + cw for some a, b, c ∈ Z.Thus the relative norm of x in this lattice is square-free.
Theorem (GKMS)If x is primitive, then this cubic lattice is unique.If a, b, c are nonzero, then x does not have a twin.
![Page 79: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/79.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
Then there exists an icube (u, v , w) with edge length msuch that x = au + bv + cw for some a, b, c ∈ Z.Thus the relative norm of x in this lattice is square-free.
Theorem (GKMS)If x is primitive, then this cubic lattice is unique.If a, b, c are nonzero, then x does not have a twin.If exactly one of them is zero, then x has two twins.
![Page 80: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/80.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
Then there exists an icube (u, v , w) with edge length msuch that x = au + bv + cw for some a, b, c ∈ Z.Thus the relative norm of x in this lattice is square-free.
Theorem (GKMS)If x is primitive, then this cubic lattice is unique.If a, b, c are nonzero, then x does not have a twin.If exactly one of them is zero, then x has two twins.If two are zero, then x has four twins,
![Page 81: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/81.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
Then there exists an icube (u, v , w) with edge length msuch that x = au + bv + cw for some a, b, c ∈ Z.Thus the relative norm of x in this lattice is square-free.
Theorem (GKMS)If x is primitive, then this cubic lattice is unique.If a, b, c are nonzero, then x does not have a twin.If exactly one of them is zero, then x has two twins.If two are zero, then x has four twins,and is contained in exactly four icubes.
![Page 82: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/82.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
Then there exists an icube (u, v , w) with edge length msuch that x = au + bv + cw for some a, b, c ∈ Z.Thus the relative norm of x in this lattice is square-free.
Theorem (GKMS)If x is primitive, then this cubic lattice is unique.If a, b, c are nonzero, then x does not have a twin.If exactly one of them is zero, then x has two twins.If two are zero, then x has four twins,and is contained in exactly four icubes.
Thus if the length of a primitive vector is an integer, thenit has 4 twins.
![Page 83: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/83.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 8 / 18
Extending primitive vectors to twins
From now on, icube means a 3-dimensional icube in Z3.
Let x ∈ Z3 whose norm is nm2, n square-free.
Then there exists an icube (u, v , w) with edge length msuch that x = au + bv + cw for some a, b, c ∈ Z.Thus the relative norm of x in this lattice is square-free.
Theorem (GKMS)If x is primitive, then this cubic lattice is unique.If a, b, c are nonzero, then x does not have a twin.If exactly one of them is zero, then x has two twins.If two are zero, then x has four twins,and is contained in exactly four icubes.
Thus if the length of a primitive vector is an integer, thenit has 4 twins. Otherwise the number of its twins is 2 or 0.
![Page 84: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/84.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 9 / 18
Constructing twins
Theorem (GKMS)If (u, v , w) is an icube and a, b ∈ Z, then (av + bw ,−bv + aw)is a twin pair.
![Page 85: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/85.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 9 / 18
Constructing twins
Theorem (GKMS)If (u, v , w) is an icube and a, b ∈ Z, then (av + bw ,−bv + aw)is a twin pair. We get all twin pairs this way.
![Page 86: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/86.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 9 / 18
Constructing twins
Theorem (GKMS)If (u, v , w) is an icube and a, b ∈ Z, then (av + bw ,−bv + aw)is a twin pair. We get all twin pairs this way. In particular,the norm of twins is the sum of two squares.
![Page 87: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/87.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 9 / 18
Constructing twins
Theorem (GKMS)If (u, v , w) is an icube and a, b ∈ Z, then (av + bw ,−bv + aw)is a twin pair. We get all twin pairs this way. In particular,the norm of twins is the sum of two squares.
Problem: this decomposition is not unique.
![Page 88: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/88.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 9 / 18
Constructing twins
Theorem (GKMS)If (u, v , w) is an icube and a, b ∈ Z, then (av + bw ,−bv + aw)is a twin pair. We get all twin pairs this way. In particular,the norm of twins is the sum of two squares.
Problem: this decomposition is not unique.
Bad example3(8,−10, 9) and 7(4, 5, 2) are twins.
![Page 89: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/89.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 9 / 18
Constructing twins
Theorem (GKMS)If (u, v , w) is an icube and a, b ∈ Z, then (av + bw ,−bv + aw)is a twin pair. We get all twin pairs this way. In particular,the norm of twins is the sum of two squares.
Problem: this decomposition is not unique.
Bad example3(8,−10, 9) and 7(4, 5, 2) are twins.Neither of them is primitive (this is the main problem).
![Page 90: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/90.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 9 / 18
Constructing twins
Theorem (GKMS)If (u, v , w) is an icube and a, b ∈ Z, then (av + bw ,−bv + aw)is a twin pair. We get all twin pairs this way. In particular,the norm of twins is the sum of two squares.
Problem: this decomposition is not unique.
Bad example3(8,−10, 9) and 7(4, 5, 2) are twins.Neither of them is primitive (this is the main problem).The “right” cubic lattice for them is given by E(0, 2, 1, 4),
![Page 91: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/91.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 9 / 18
Constructing twins
Theorem (GKMS)If (u, v , w) is an icube and a, b ∈ Z, then (av + bw ,−bv + aw)is a twin pair. We get all twin pairs this way. In particular,the norm of twins is the sum of two squares.
Problem: this decomposition is not unique.
Bad example3(8,−10, 9) and 7(4, 5, 2) are twins.Neither of them is primitive (this is the main problem).The “right” cubic lattice for them is given by E(0, 2, 1, 4),that is u = (−13, 4, 16), v = (4,−19, 8), w = (16, 8, 11)
![Page 92: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/92.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 9 / 18
Constructing twins
Theorem (GKMS)If (u, v , w) is an icube and a, b ∈ Z, then (av + bw ,−bv + aw)is a twin pair. We get all twin pairs this way. In particular,the norm of twins is the sum of two squares.
Problem: this decomposition is not unique.
Bad example3(8,−10, 9) and 7(4, 5, 2) are twins.Neither of them is primitive (this is the main problem).The “right” cubic lattice for them is given by E(0, 2, 1, 4),that is u = (−13, 4, 16), v = (4,−19, 8), w = (16, 8, 11)with a = 2 and b = 1.
![Page 93: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/93.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 9 / 18
Constructing twins
Theorem (GKMS)If (u, v , w) is an icube and a, b ∈ Z, then (av + bw ,−bv + aw)is a twin pair. We get all twin pairs this way. In particular,the norm of twins is the sum of two squares.
Problem: this decomposition is not unique.
Bad example3(8,−10, 9) and 7(4, 5, 2) are twins.Neither of them is primitive (this is the main problem).The “right” cubic lattice for them is given by E(0, 2, 1, 4),that is u = (−13, 4, 16), v = (4,−19, 8), w = (16, 8, 11)with a = 2 and b = 1.
How to “foresee” the “divisors” of a non-primitive au + bv + cw?
![Page 94: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/94.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M.
![Page 95: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/95.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
![Page 96: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/96.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(where pr and qs are primes
![Page 97: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/97.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(
pr ≡ 1 (4),
(where pr and qs are primes
![Page 98: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/98.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(
pr ≡ 1 (4), qs ≡ −1 (4))
(where pr and qs are primes
![Page 99: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/99.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(
pr ≡ 1 (4), qs ≡ −1 (4))
(where pr and qs are primes and λr , µs > 0),
![Page 100: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/100.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(
pr ≡ 1 (4), qs ≡ −1 (4))
(where pr and qs are primes and λr , µs > 0), then
T(M) = 24m∏
r=1
g(pλrr )
ℓ∏
s=1
h(qµss ) ,
![Page 101: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/101.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(
pr ≡ 1 (4), qs ≡ −1 (4))
(where pr and qs are primes and λr , µs > 0), then
T(M) = 24m∏
r=1
g(pλrr )
ℓ∏
s=1
h(qµss ) ,
where
g(p2λ) = σ(pλ) + σ(pλ−1) ,
![Page 102: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/102.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(
pr ≡ 1 (4), qs ≡ −1 (4))
(where pr and qs are primes and λr , µs > 0), then
T(M) = 24m∏
r=1
g(pλrr )
ℓ∏
s=1
h(qµss ) ,
where
g(p2λ) = σ(pλ) + σ(pλ−1) , g(p2λ+1) = 2σ(pλ) ,
![Page 103: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/103.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(
pr ≡ 1 (4), qs ≡ −1 (4))
(where pr and qs are primes and λr , µs > 0), then
T(M) = 24m∏
r=1
g(pλrr )
ℓ∏
s=1
h(qµss ) ,
where
g(p2λ) = σ(pλ) + σ(pλ−1) , g(p2λ+1) = 2σ(pλ) ,
h(q2µ) = σ(qµ) + σ(qµ−1) ,
![Page 104: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/104.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(
pr ≡ 1 (4), qs ≡ −1 (4))
(where pr and qs are primes and λr , µs > 0), then
T(M) = 24m∏
r=1
g(pλrr )
ℓ∏
s=1
h(qµss ) ,
where
g(p2λ) = σ(pλ) + σ(pλ−1) , g(p2λ+1) = 2σ(pλ) ,
h(q2µ) = σ(qµ) + σ(qµ−1) , h(q2µ+1) = 0 .
![Page 105: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/105.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(
pr ≡ 1 (4), qs ≡ −1 (4))
(where pr and qs are primes and λr , µs > 0), then
T(M) = 24m∏
r=1
g(pλrr )
ℓ∏
s=1
h(qµss ) ,
where
g(p2λ) = σ(pλ) + σ(pλ−1) , g(p2λ+1) = 2σ(pλ) ,
h(q2µ) = σ(qµ) + σ(qµ−1) , h(q2µ+1) = 0 .
In particular, T(M)/24 is a multiplicative function.
![Page 106: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/106.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 10 / 18
Counting twins
Theorem (GKMS)Denote by T(M) the number of twin pairs whose norm(length squared) is M. Suppose that
M = 2κpλ11 . . . pλm
m qµ11 . . . qµℓ
ℓ
(
pr ≡ 1 (4), qs ≡ −1 (4))
(where pr and qs are primes and λr , µs > 0), then
T(M) = 24m∏
r=1
g(pλrr )
ℓ∏
s=1
h(qµss ) ,
where
g(p2λ) = σ(pλ) + σ(pλ−1) , g(p2λ+1) = 2σ(pλ) ,
h(q2µ) = σ(qµ) + σ(qµ−1) , h(q2µ+1) = 0 .
In particular, T(M)/24 is a multiplicative function.
Proof: using integral quaternions.
![Page 107: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/107.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
![Page 108: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/108.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk
![Page 109: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/109.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,
![Page 110: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/110.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k
![Page 111: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/111.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1
![Page 112: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/112.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).
![Page 113: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/113.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3
![Page 114: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/114.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3 whose matrix is E(m, n, p, q).
![Page 115: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/115.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3 whose matrix is E(m, n, p, q).Conversely, every rotation
![Page 116: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/116.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3 whose matrix is E(m, n, p, q).Conversely, every rotation (element of the group SO(R3))
![Page 117: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/117.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3 whose matrix is E(m, n, p, q).Conversely, every rotation (element of the group SO(R3))can be obtained in such a way,
![Page 118: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/118.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3 whose matrix is E(m, n, p, q).Conversely, every rotation (element of the group SO(R3))can be obtained in such a way, and α is unique up to sign.
![Page 119: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/119.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3 whose matrix is E(m, n, p, q).Conversely, every rotation (element of the group SO(R3))can be obtained in such a way, and α is unique up to sign.
ExampleLet α = 2i + j + 4k ,
![Page 120: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/120.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3 whose matrix is E(m, n, p, q).Conversely, every rotation (element of the group SO(R3))can be obtained in such a way, and α is unique up to sign.
ExampleLet α = 2i + j + 4k , its norm is 21.
![Page 121: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/121.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3 whose matrix is E(m, n, p, q).Conversely, every rotation (element of the group SO(R3))can be obtained in such a way, and α is unique up to sign.
ExampleLet α = 2i + j + 4k , its norm is 21.Then θ 7→ αθα is a dilated rotation.
![Page 122: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/122.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3 whose matrix is E(m, n, p, q).Conversely, every rotation (element of the group SO(R3))can be obtained in such a way, and α is unique up to sign.
ExampleLet α = 2i + j + 4k , its norm is 21.Then θ 7→ αθα is a dilated rotation.It transforms the “planar” twin pair (2j + k ,−j + 2k) (norm 5)
![Page 123: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/123.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 11 / 18
Geometry and quaternions
Well-known in geometry
Identify (x1, x2, x3) ∈ R3 and the pure quaternion x1i + x2j + x3k .
Let α = m + ni + pj + qk with N(α) = m2 + n2 + p2 + q2 = 1,and for θ = x1i + x2j + x3k let E(α) : θ 7→ αθα−1(= αθα).Then E(α) yields a rotation of R
3 whose matrix is E(m, n, p, q).Conversely, every rotation (element of the group SO(R3))can be obtained in such a way, and α is unique up to sign.
ExampleLet α = 2i + j + 4k , its norm is 21.Then θ 7→ αθα is a dilated rotation.It transforms the “planar” twin pair (2j + k ,−j + 2k) (norm 5)to the twin pair (24i − 30j + 27k , 28i + 35j + 14k) (norm 212 · 5).
![Page 124: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/124.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions,
![Page 125: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/125.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d are
![Page 126: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/126.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers,
![Page 127: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/127.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.
![Page 128: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/128.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization
![Page 129: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/129.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).
![Page 130: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/130.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).E has 24 units
![Page 131: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/131.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).E has 24 units (σ = (1 + i + j + k)/2 is one).
![Page 132: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/132.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).E has 24 units (σ = (1 + i + j + k)/2 is one).The irreducible elements of E are the ones with prime norm.
![Page 133: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/133.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).E has 24 units (σ = (1 + i + j + k)/2 is one).The irreducible elements of E are the ones with prime norm.There are 24(p + 1) such elements whose norm is p > 2,
![Page 134: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/134.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).E has 24 units (σ = (1 + i + j + k)/2 is one).The irreducible elements of E are the ones with prime norm.There are 24(p + 1) such elements whose norm is p > 2,and the elements with norm 2 are the 24 associates of 1 + i .
![Page 135: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/135.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).E has 24 units (σ = (1 + i + j + k)/2 is one).The irreducible elements of E are the ones with prime norm.There are 24(p + 1) such elements whose norm is p > 2,and the elements with norm 2 are the 24 associates of 1 + i .
It is usually sufficient to use the following for uniqueness:
![Page 136: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/136.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).E has 24 units (σ = (1 + i + j + k)/2 is one).The irreducible elements of E are the ones with prime norm.There are 24(p + 1) such elements whose norm is p > 2,and the elements with norm 2 are the 24 associates of 1 + i .
It is usually sufficient to use the following for uniqueness:
If a prime p divides N(α)
![Page 137: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/137.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).E has 24 units (σ = (1 + i + j + k)/2 is one).The irreducible elements of E are the ones with prime norm.There are 24(p + 1) such elements whose norm is p > 2,and the elements with norm 2 are the 24 associates of 1 + i .
It is usually sufficient to use the following for uniqueness:
If a prime p divides N(α) but does not divide α,
![Page 138: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/138.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).E has 24 units (σ = (1 + i + j + k)/2 is one).The irreducible elements of E are the ones with prime norm.There are 24(p + 1) such elements whose norm is p > 2,and the elements with norm 2 are the 24 associates of 1 + i .
It is usually sufficient to use the following for uniqueness:
If a prime p divides N(α) but does not divide α,then α = πα′ for some π with norm p,
![Page 139: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/139.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 12 / 18
Hurwitz integral quaternions
Well-known in algebraLet E denote the ring of Hurwitz-quaternions, that is,quaternions a + bi + cj + dk such that a, b, c, d areeither all integers, or all of them is the half of an odd integer.Then E has “unique” factorization (it is right Euclidean).E has 24 units (σ = (1 + i + j + k)/2 is one).The irreducible elements of E are the ones with prime norm.There are 24(p + 1) such elements whose norm is p > 2,and the elements with norm 2 are the 24 associates of 1 + i .
It is usually sufficient to use the following for uniqueness:
If a prime p divides N(α) but does not divide α,then α = πα′ for some π with norm p,and π is unique up to right association.
![Page 140: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/140.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,
![Page 141: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/141.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).
![Page 142: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/142.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
![Page 143: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/143.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p,
![Page 144: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/144.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,
![Page 145: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/145.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α),
![Page 146: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/146.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α), and there is an integer h
![Page 147: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/147.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α), and there is an integer hand a right divisor π of α
![Page 148: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/148.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α), and there is an integer hand a right divisor π of α such that N(π) = p
![Page 149: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/149.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α), and there is an integer hand a right divisor π of α such that N(π) = p and π | h + β.
![Page 150: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/150.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α), and there is an integer hand a right divisor π of α such that N(π) = p and π | h + β.
Let s(M) denote the number of vectors with norm M.
![Page 151: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/151.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α), and there is an integer hand a right divisor π of α such that N(π) = p and π | h + β.
Let s(M) denote the number of vectors with norm M.This lemma reduces its computation to the square-free case.
![Page 152: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/152.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α), and there is an integer hand a right divisor π of α such that N(π) = p and π | h + β.
Let s(M) denote the number of vectors with norm M.This lemma reduces its computation to the square-free case.
Corollary used for twin-completeness laterFor every primitive pure β ∈ E
![Page 153: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/153.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α), and there is an integer hand a right divisor π of α such that N(π) = p and π | h + β.
Let s(M) denote the number of vectors with norm M.This lemma reduces its computation to the square-free case.
Corollary used for twin-completeness laterFor every primitive pure β ∈ E and m > 0
![Page 154: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/154.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α), and there is an integer hand a right divisor π of α such that N(π) = p and π | h + β.
Let s(M) denote the number of vectors with norm M.This lemma reduces its computation to the square-free case.
Corollary used for twin-completeness laterFor every primitive pure β ∈ E and m > 0there is an α ∈ E with norm m
![Page 155: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/155.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 13 / 18
Decomposing single vectors
Every pure quaternion θ ∈ E can be written as αβ α,where N(β) is the square-free part of N(θ) (and α ∈ E).If θ is primitive, then α is unique up to right associates.
Lemma (GKMS)If αβ α is divisible by an odd prime p, but α and β is not,then p | N(α), and there is an integer hand a right divisor π of α such that N(π) = p and π | h + β.
Let s(M) denote the number of vectors with norm M.This lemma reduces its computation to the square-free case.
Corollary used for twin-completeness laterFor every primitive pure β ∈ E and m > 0there is an α ∈ E with norm m such that αβ α is primitive.
![Page 156: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/156.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
![Page 157: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/157.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff
![Page 158: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/158.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.
![Page 159: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/159.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E
![Page 160: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/160.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).
![Page 161: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/161.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α
![Page 162: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/162.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α and η = αzk α are obviously twins.
![Page 163: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/163.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α and η = αzk α are obviously twins.We say that (θ, η) is parameterized by (α, z) ∈ E×G.
![Page 164: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/164.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α and η = αzk α are obviously twins.We say that (θ, η) is parameterized by (α, z) ∈ E×G.
Theorem (GKMS)Each twin pair is parametrized by some pair in E×G,
![Page 165: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/165.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α and η = αzk α are obviously twins.We say that (θ, η) is parameterized by (α, z) ∈ E×G.
Theorem (GKMS)Each twin pair is parametrized by some pair in E×G,where the second component is square-free in G.
![Page 166: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/166.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α and η = αzk α are obviously twins.We say that (θ, η) is parameterized by (α, z) ∈ E×G.
Theorem (GKMS)Each twin pair is parametrized by some pair in E×G,where the second component is square-free in G.Such (α1, z1), (α2, z2) ∈ E×G yield the same twin pairiff there exists a unit ρ ∈ G
![Page 167: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/167.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α and η = αzk α are obviously twins.We say that (θ, η) is parameterized by (α, z) ∈ E×G.
Theorem (GKMS)Each twin pair is parametrized by some pair in E×G,where the second component is square-free in G.Such (α1, z1), (α2, z2) ∈ E×G yield the same twin pairiff there exists a unit ρ ∈ G (that is, an element of {±1,±i})
![Page 168: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/168.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α and η = αzk α are obviously twins.We say that (θ, η) is parameterized by (α, z) ∈ E×G.
Theorem (GKMS)Each twin pair is parametrized by some pair in E×G,where the second component is square-free in G.Such (α1, z1), (α2, z2) ∈ E×G yield the same twin pairiff there exists a unit ρ ∈ G (that is, an element of {±1,±i})such that α2 = α1ρ
![Page 169: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/169.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α and η = αzk α are obviously twins.We say that (θ, η) is parameterized by (α, z) ∈ E×G.
Theorem (GKMS)Each twin pair is parametrized by some pair in E×G,where the second component is square-free in G.Such (α1, z1), (α2, z2) ∈ E×G yield the same twin pairiff there exists a unit ρ ∈ G (that is, an element of {±1,±i})such that α2 = α1ρ and z1 = ρ2z2.
![Page 170: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/170.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α and η = αzk α are obviously twins.We say that (θ, η) is parameterized by (α, z) ∈ E×G.
Theorem (GKMS)Each twin pair is parametrized by some pair in E×G,where the second component is square-free in G.Such (α1, z1), (α2, z2) ∈ E×G yield the same twin pairiff there exists a unit ρ ∈ G (that is, an element of {±1,±i})such that α2 = α1ρ and z1 = ρ2z2.
We get an icube exactly when z is real
![Page 171: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/171.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 14 / 18
Constructing twin pairs
For u, v ∈ Z3 let θ, η be the corresponding pure quaternions.
Then u ⊥ v iff θη is also a pure quaternion.Let α ∈ E and z ∈ G (Gaussian integers).Then θ = αzj α and η = αzk α are obviously twins.We say that (θ, η) is parameterized by (α, z) ∈ E×G.
Theorem (GKMS)Each twin pair is parametrized by some pair in E×G,where the second component is square-free in G.Such (α1, z1), (α2, z2) ∈ E×G yield the same twin pairiff there exists a unit ρ ∈ G (that is, an element of {±1,±i})such that α2 = α1ρ and z1 = ρ2z2.
We get an icube exactly when z is real or pure imaginary.
![Page 172: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/172.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 15 / 18
Twin-complete numbers
RecallA vector can be put into an icube iff its norm is a square.
![Page 173: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/173.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 15 / 18
Twin-complete numbers
RecallA vector can be put into an icube iff its norm is a square.
Extension: Which non-primitive vectors have a twin?
![Page 174: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/174.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 15 / 18
Twin-complete numbers
RecallA vector can be put into an icube iff its norm is a square.
Extension: Which non-primitive vectors have a twin?
Necessary: The norm must be the sum of two squares.
![Page 175: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/175.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 15 / 18
Twin-complete numbers
RecallA vector can be put into an icube iff its norm is a square.
Extension: Which non-primitive vectors have a twin?
Necessary: The norm must be the sum of two squares.
Easier ProblemCharacterize those numbers M such thatevery integral vector of norm M has a twin.
![Page 176: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/176.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 15 / 18
Twin-complete numbers
RecallA vector can be put into an icube iff its norm is a square.
Extension: Which non-primitive vectors have a twin?
Necessary: The norm must be the sum of two squares.
Easier ProblemCharacterize those numbers M such thatevery integral vector of norm M has a twin.
Exclude those M for which there is no vector of norm M
![Page 177: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/177.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 15 / 18
Twin-complete numbers
RecallA vector can be put into an icube iff its norm is a square.
Extension: Which non-primitive vectors have a twin?
Necessary: The norm must be the sum of two squares.
Easier ProblemCharacterize those numbers M such thatevery integral vector of norm M has a twin.
Exclude those M for which there is no vector of norm M(that is, numbers M of the form 4n(8k + 7)).
![Page 178: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/178.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 15 / 18
Twin-complete numbers
RecallA vector can be put into an icube iff its norm is a square.
Extension: Which non-primitive vectors have a twin?
Necessary: The norm must be the sum of two squares.
Easier ProblemCharacterize those numbers M such thatevery integral vector of norm M has a twin.
Exclude those M for which there is no vector of norm M(that is, numbers M of the form 4n(8k + 7)).
Such numbers M are called twin-complete.
![Page 179: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/179.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 16 / 18
Characterizing twin-completeness
Theorem (GKMS)A positive integer is twin-complete if and only if
![Page 180: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/180.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 16 / 18
Characterizing twin-completeness
Theorem (GKMS)A positive integer is twin-complete if and only ifits squre-free part can be written as a sum of two squares,
![Page 181: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/181.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 16 / 18
Characterizing twin-completeness
Theorem (GKMS)A positive integer is twin-complete if and only ifits squre-free part can be written as a sum of two squares,but not as a sum of three positive squares.
![Page 182: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/182.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 16 / 18
Characterizing twin-completeness
Theorem (GKMS)A positive integer is twin-complete if and only ifits squre-free part can be written as a sum of two squares,but not as a sum of three positive squares.
The proof uses the machinery built above.
![Page 183: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/183.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 16 / 18
Characterizing twin-completeness
Theorem (GKMS)A positive integer is twin-complete if and only ifits squre-free part can be written as a sum of two squares,but not as a sum of three positive squares.
The proof uses the machinery built above.
Famous conjecture in number theoryThe complete list of positive square-free integers
![Page 184: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/184.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 16 / 18
Characterizing twin-completeness
Theorem (GKMS)A positive integer is twin-complete if and only ifits squre-free part can be written as a sum of two squares,but not as a sum of three positive squares.
The proof uses the machinery built above.
Famous conjecture in number theoryThe complete list of positive square-free integersthat can be written as a sum of two squares,
![Page 185: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/185.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 16 / 18
Characterizing twin-completeness
Theorem (GKMS)A positive integer is twin-complete if and only ifits squre-free part can be written as a sum of two squares,but not as a sum of three positive squares.
The proof uses the machinery built above.
Famous conjecture in number theoryThe complete list of positive square-free integersthat can be written as a sum of two squares,but not as a sum of three positive squares,
![Page 186: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/186.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 16 / 18
Characterizing twin-completeness
Theorem (GKMS)A positive integer is twin-complete if and only ifits squre-free part can be written as a sum of two squares,but not as a sum of three positive squares.
The proof uses the machinery built above.
Famous conjecture in number theoryThe complete list of positive square-free integersthat can be written as a sum of two squares,but not as a sum of three positive squares,is the following: 1, 2, 5, 10, 13, 37, 58, 85, 130.
![Page 187: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/187.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 16 / 18
Characterizing twin-completeness
Theorem (GKMS)A positive integer is twin-complete if and only ifits squre-free part can be written as a sum of two squares,but not as a sum of three positive squares.
The proof uses the machinery built above.
Famous conjecture in number theoryThe complete list of positive square-free integersthat can be written as a sum of two squares,but not as a sum of three positive squares,is the following: 1, 2, 5, 10, 13, 37, 58, 85, 130.
If true, then d2, 2d2, 5d2, 10d2, 13d2, 37d2, 58d2, 85d2, 130d2
are exactly the twin-complete numbers.
![Page 188: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/188.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m,
![Page 189: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/189.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m, if m = x2 + ny2, (x2, ny2) = 1,x , y ≥ 0 has a unique solution,
![Page 190: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/190.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m, if m = x2 + ny2, (x2, ny2) = 1,x , y ≥ 0 has a unique solution, then m is a prime power,or twice a prime power.
![Page 191: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/191.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m, if m = x2 + ny2, (x2, ny2) = 1,x , y ≥ 0 has a unique solution, then m is a prime power,or twice a prime power.
Euler’s conjecture: his list of 65 numeri idonei is complete.
![Page 192: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/192.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m, if m = x2 + ny2, (x2, ny2) = 1,x , y ≥ 0 has a unique solution, then m is a prime power,or twice a prime power.
Euler’s conjecture: his list of 65 numeri idonei is complete.Every number in the conjecture above is a numerus idoneus.
![Page 193: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/193.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m, if m = x2 + ny2, (x2, ny2) = 1,x , y ≥ 0 has a unique solution, then m is a prime power,or twice a prime power.
Euler’s conjecture: his list of 65 numeri idonei is complete.Every number in the conjecture above is a numerus idoneus.Relationship: positive solutions of xy + xz + yz = n.
![Page 194: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/194.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m, if m = x2 + ny2, (x2, ny2) = 1,x , y ≥ 0 has a unique solution, then m is a prime power,or twice a prime power.
Euler’s conjecture: his list of 65 numeri idonei is complete.Every number in the conjecture above is a numerus idoneus.Relationship: positive solutions of xy + xz + yz = n.S. Chowla: there are only finitely many numeri idonei.
![Page 195: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/195.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m, if m = x2 + ny2, (x2, ny2) = 1,x , y ≥ 0 has a unique solution, then m is a prime power,or twice a prime power.
Euler’s conjecture: his list of 65 numeri idonei is complete.Every number in the conjecture above is a numerus idoneus.Relationship: positive solutions of xy + xz + yz = n.S. Chowla: there are only finitely many numeri idonei.P. J. Weinberger: At most one square-free number can bemissing from Euler’s list,
![Page 196: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/196.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m, if m = x2 + ny2, (x2, ny2) = 1,x , y ≥ 0 has a unique solution, then m is a prime power,or twice a prime power.
Euler’s conjecture: his list of 65 numeri idonei is complete.Every number in the conjecture above is a numerus idoneus.Relationship: positive solutions of xy + xz + yz = n.S. Chowla: there are only finitely many numeri idonei.P. J. Weinberger: At most one square-free number can bemissing from Euler’s list, and it is greater than 2 · 1011
![Page 197: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/197.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m, if m = x2 + ny2, (x2, ny2) = 1,x , y ≥ 0 has a unique solution, then m is a prime power,or twice a prime power.
Euler’s conjecture: his list of 65 numeri idonei is complete.Every number in the conjecture above is a numerus idoneus.Relationship: positive solutions of xy + xz + yz = n.S. Chowla: there are only finitely many numeri idonei.P. J. Weinberger: At most one square-free number can bemissing from Euler’s list, and it is greater than 2 · 1011
(the largest number on Euler’s list is 1848).
![Page 198: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/198.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 17 / 18
Euler’s numeri idonei
Euler defined a numerus idoneus to be an integer n such that,for any positive integer m, if m = x2 + ny2, (x2, ny2) = 1,x , y ≥ 0 has a unique solution, then m is a prime power,or twice a prime power.
Euler’s conjecture: his list of 65 numeri idonei is complete.Every number in the conjecture above is a numerus idoneus.Relationship: positive solutions of xy + xz + yz = n.S. Chowla: there are only finitely many numeri idonei.P. J. Weinberger: At most one square-free number can bemissing from Euler’s list, and it is greater than 2 · 1011
(the largest number on Euler’s list is 1848).If the Generalized Riemann Hypothesis holds, then the possiblemissing tenth number on the twin-complete list is odd.
![Page 199: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/199.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.
![Page 200: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/200.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.
![Page 201: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/201.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.So suppose that the dimension is odd.
![Page 202: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/202.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.So suppose that the dimension is odd. What are thepossible norms of twins?
![Page 203: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/203.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.So suppose that the dimension is odd. What are thepossible norms of twins? What about twin-completeness?
![Page 204: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/204.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.So suppose that the dimension is odd. What are thepossible norms of twins? What about twin-completeness?Call a vector odd, if each of its components is odd.
![Page 205: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/205.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.So suppose that the dimension is odd. What are thepossible norms of twins? What about twin-completeness?Call a vector odd, if each of its components is odd.Obvious: In odd dimensions, odd vectors cannot have a twin.
![Page 206: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/206.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.So suppose that the dimension is odd. What are thepossible norms of twins? What about twin-completeness?Call a vector odd, if each of its components is odd.Obvious: In odd dimensions, odd vectors cannot have a twin.
Conjecture (may be easy!)Suppose that the dimension n ≥ 5 is odd.Then every non-odd vector has a twin.
![Page 207: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/207.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.So suppose that the dimension is odd. What are thepossible norms of twins? What about twin-completeness?Call a vector odd, if each of its components is odd.Obvious: In odd dimensions, odd vectors cannot have a twin.
Conjecture (may be easy!)Suppose that the dimension n ≥ 5 is odd.Then every non-odd vector has a twin.
Obvious: Every odd vector of norm M satisfies that M ≡ n (8).
![Page 208: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/208.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.So suppose that the dimension is odd. What are thepossible norms of twins? What about twin-completeness?Call a vector odd, if each of its components is odd.Obvious: In odd dimensions, odd vectors cannot have a twin.
Conjecture (may be easy!)Suppose that the dimension n ≥ 5 is odd.Then every non-odd vector has a twin.
Obvious: Every odd vector of norm M satisfies that M ≡ n (8).
Weaker conjectureSuppose that the dimension n ≥ 5 is odd
![Page 209: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/209.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.So suppose that the dimension is odd. What are thepossible norms of twins? What about twin-completeness?Call a vector odd, if each of its components is odd.Obvious: In odd dimensions, odd vectors cannot have a twin.
Conjecture (may be easy!)Suppose that the dimension n ≥ 5 is odd.Then every non-odd vector has a twin.
Obvious: Every odd vector of norm M satisfies that M ≡ n (8).
Weaker conjectureSuppose that the dimension n ≥ 5 is odd and M 6≡ n (8).
![Page 210: An application of integral quaternionsewkiss.web.elte.hu/wp/wordpress/wp-content/uploads/2014/... · 2014-11-18 · An application of integral quaternions Goswick, Kiss, Moussong,](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0dc6b47e708231d43c077d/html5/thumbnails/210.jpg)
An application of integral quaternions Goswick, Kiss, Moussong, Simányi 18 / 18
Problems in higher dimensions
Study construction, counting, extension for general icubes.Obvious: In even dimensions n, every vector has a twin.So suppose that the dimension is odd. What are thepossible norms of twins? What about twin-completeness?Call a vector odd, if each of its components is odd.Obvious: In odd dimensions, odd vectors cannot have a twin.
Conjecture (may be easy!)Suppose that the dimension n ≥ 5 is odd.Then every non-odd vector has a twin.
Obvious: Every odd vector of norm M satisfies that M ≡ n (8).
Weaker conjectureSuppose that the dimension n ≥ 5 is odd and M 6≡ n (8).Then every vector of norm M has a twin.