PascGalois Activities

for aNumber Theory ClassKurt Ludwick

Salisbury UniversitySalisbury, MD

http://faculty.salisbury.edu/~keludwick

Developed by Kathleen M. Shannon & Michael J. Bardzell Salisbury University, Salisbury, MD

Support provided by The National Science Foundation award #'s DUE-0087644 and

DUE-0339477-and-

The Richard A. Henson endowment for the School of Science at Salisbury University

PascGaloisVisualization Software for Abstract Algebra

http://www.pascgalois.org

PascGaloishttp://www.pascgalois.org

Applications in a Number Theory class?

• Main idea: Pascal’s Triangle modulo n ( 1-dimensional finite automata)

• Primary use: Abstract Algebra classes ( Visualization of subgroups, cosets, etc.)

A few class objectives:• Properties of modular arithmetic• Properties of binomial coefficients • Inductive reasoning

• Observing a pattern• Clearly stating a hypothesis

• Proof • Significance of prime factorization

A few PascGalois examples…..

Pascal’s Triangle modulo 2 – rows 0 - 64

Even numbers: redOdd numbers: black

A few examples…..A few examples…..

Pascal’s Triangle modulo 5 – rows 0 - 50Colors correspond to remainders

Notice “inverted” red triangles, as were also seen in the modulo 2 triangle

A few examples…..A few examples…..

Pascal’s Triangle modulo 12 – rows 0 - 72

Activity #1 – Inverted triangles

Discovery activity (ideally) – best suited as interactive assignment in a computer lab (can also work as an out-of-class assignment, with detailed instructions)

Notice the solid triangles with side length at least 3within Pascal’s Triangle (modulo 2).

What do we observe about them?

• They are all red

• They are all “upside down” (Longest edge is at the top)

• Their sizes vary throughout the interior of Pascal’s Triangle (modulo 2)

These characteristics can be seen under other moduli as well…..

Notice the solid triangles with side length at least 3within Pascal’s Triangle (modulo 5).

What do we observe about them?

• They are all red

• They are all “upside down” (Longest edge is at the top)

• Their sizes vary throughout the interior of Pascal’s Triangle (modulo 5)

These characteristics can be seen under other moduli as well…..

Activity #1 – Inverted triangles

Activity #1 – Inverted triangles

Questions:

1. Are the solid triangles always inverted?

2. Are the solid triangles always red?

3. Find the size (defined as length of the top row) of each triangle as a function of the row number in which its top row appears

Activity #1 – Inverted triangles

Questions:

1. Are the solid triangles always inverted?

Suppose not… then, the following must occur somewhere within Pascal’s Triangle (modulo n)for some X, 0 < X < n-1:

…where none of the entries labeled “?” may be equal to X

We can see that certain of the “?” entries must be 0 (implying X is not 0)……

Activity #1 – Inverted triangles

Questions:

1. Are the solid triangles always inverted?

Suppose not… then, the following must occur somewhere within Pascal’s Triangle (modulo n)for some X, 0 < X < n-1:

Thus, by contradiction (and using properties of modular arithmetic), no “right-side-up” triangles of size 3 (or greater) can occur.

Activity #1 – Inverted triangles

Questions:

2. Are the solid triangles always red?

Yes, by a similar argument… to have an inverted triangle of a single color, X, it would be necessary to have

which implies X = 0 , or red.

(The standard coloring scheme in PascGalois is to have red designate the zero remainder.This can be customized, of course!)

),n (mod XXX

Activity #1 – Inverted triangles

Questions:

3. Find the size (defined as length of the top row) of each triangle as a function of the row number in which its top row appears

Size = 3….

Row 4

Row 12

Row 20

Row 28

etc.

Activity #1 – Inverted triangles

Questions:

3. Find the size (defined as length of the top row) of each triangle as a function of the row number in which its top row appears

Size = 7….

Row 8

Row 24

Row 40

Row 56

Activity #1 – Inverted triangles

Questions:

3. Find the size (defined as length of the top row) of each triangle as a function of the row number in which its top row appears

Size = 15….

Row 16

Row 48

Next: 80, 112, 144, etc…

Activity #1 – Inverted triangles

Questions:

3. Find the size (defined as length of the top row) of each triangle as a function of the row number in which its top row appears

Size = 31….

Row 32

…..next?

Activity #1 – Inverted triangles

Questions:

3. Find the size (defined as length of the top row) of each triangle as a function of the row number in which its top row appears

To answer this question completely, one must usethe prime factorization of the row number.

In Pascal’s Triangle (modulo 2):• Size 3 triangles begin in rows numbered 2

2M,

where M is a product of primes not equal to 2 (same meaning for “M” throughout…..)

• Size 7 triangles begin in rows numbered 23M

• Size 15 triangles begin in rows numbered 24M

…and so on… in general, within Pascal’s Triangle (modulo 2), the size of a solid red triangle starting on a given row will be 2

k-1, where 2

k is the greatest power of 2

that divides the row number

Activity #1 – Inverted triangles

Questions:

3. Find the size (defined as length of the top row) of each triangle as a function of the row number in which its top row appears

Generalizing to Pascal’s Triangle (modulo p), for prime p:

• Size p-1 triangles begin in rows numbered pM, where M is a product of primes not equal to p

• Size p2-1 triangles begin in rows numbered p

2M

…within Pascal’s Triangle (modulo p), p an odd prime, the size of a solid red triangle will be p

k-1, where p

k is the

greatest power of p that divides the row number

To come up with this solution, students must get used to thinking about integers in terms of their prime factorization.

Activity #1 – Inverted triangles

Questions:

3. Find the size (defined as length of the top row) of each triangle as a function of the row number in which its top row appears

Example: Pascal’s Triangle (modulo 5):

Red triangles of size 4 begin on rows 5, 10, 15, and 20.

A red triangle of size 24 begins on row 25.

More triangles of size 4 begin on rows 30, 35, 40 and 45…..

Guess what happens on row 50?

Activity #1 – Inverted triangles

Summary:• Gives students experience working with the PascGalois software

• Provides a few “easy” proofs involving properties of modular arithmetic

• Introduces (or reinforces) the idea of thinking of the natural numbers in terms of their prime factorizations

Activity #2 – Lucas Correspondence Theorem

Instructions:

• Choose a prime number, p. Use PascGalois to generate Pascal’s Triangle modulo p.

• Choose a row in this triangle. Let r denote the row number you choose.

• Write out each of the following in base p:o The row number, ro From row r, the horizontal position of each non-red (non-zero) entry

As an example, we will consider row r=32of Pascal’s Triangle modulo 5. So, r=1125.

The non-zero locations in this row are: 0, 1, 2, 5, 6, 7, 25, 26, 27, 30, 31 and 32.

Activity #2 – Lucas Correspondence Theorem

Observation (after a few examples):The k

th position in row r is nonzero (mod p) iff each digit of k is less than or equal to the corresponding base p digit of r.

This is an observation in the direction of what is known as the Lucas Correspondence Theorem…..

Activity #2 – Lucas Correspondence Theorem

The Lucas Correspondence Theorem:

.0 , ,0 , prprrpkpkk ii

i

iii

i

i

).(mod pk

r

k

r

i i

i

Let p be prime, and let k, r be positive integers with base p digits ki, ri, respectively.

That is,

Then,

Notice: iff ri < ki, which is why

an entry in the kth position is zero (mod p) iff at least one of its base p digits is greater than the corresponding digits of the row number, r.

0

i

i

k

r

Activity #2 – Lucas Correspondence Theorem

Following the same example, we have….

Activity #2 – Lucas Correspondence Theorem

Following the same example, we have….

Activity #2 – Lucas Correspondence Theorem

Following the same example, we have….

Note: for any other value of k, one of the three factors (and thus the product) in the right-hand column is zero, corresponding to a binomial coefficient that is congruent to 0 (mod 5), as per Lucas’s Theorem.

Activity #2 – Lucas Correspondence Theorem

Example: Pascal’s Triangle (mod 7), row 23

Pascal’s Triangle (mod 7)Rows 0-27

r = 23 = 327

Nonzeros: k = 0, 1, 2, 7, 8, 9, 14, 15, 16, 21, 22, 23

Developed by Kathleen M. Shannon & Michael J. Bardzell Salisbury University, Salisbury, MD

Support provided by The National Science Foundation award #'s DUE-0087644 and

DUE-0339477-and-

The Richard A. Henson endowment for the School of Science at Salisbury University

PascGaloisVisualization Software for Abstract Algebra

http://www.pascgalois.org

THANK YOU!