On zero-sum problems
description
Transcript of On zero-sum problems
![Page 1: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/1.jpg)
Joseph DiMuro, 12/6/11
![Page 2: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/2.jpg)
An introductory challenge
![Page 3: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/3.jpg)
Calculate Reset
![Page 4: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/4.jpg)
A zero-sum problem
![Page 5: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/5.jpg)
A zero-sum problem
![Page 6: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/6.jpg)
![Page 7: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/7.jpg)
![Page 8: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/8.jpg)
![Page 9: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/9.jpg)
Outline
Definitions from abstract algebra The zero-sum problem for general
groups Variations of the zero-sum problem An application of zero-sum problems
![Page 10: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/10.jpg)
Finite abelian groups
![Page 11: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/11.jpg)
Finite abelian groups
![Page 12: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/12.jpg)
Other finite abelian groups
![Page 13: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/13.jpg)
The Davenport constant
![Page 14: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/14.jpg)
Isomorphic abelian groups
![Page 15: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/15.jpg)
Isomorphic abelian groups
![Page 16: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/16.jpg)
Isomorphic abelian groups
![Page 17: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/17.jpg)
Isomorphic abelian groups
Generators:
![Page 18: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/18.jpg)
Isomorphic abelian groups
![Page 19: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/19.jpg)
Isomorphic abelian groups
![Page 20: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/20.jpg)
Isomorphic abelian groups
![Page 21: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/21.jpg)
Isomorphic abelian groups
Finite abelian groups may always be written as direct products of groups of prime power order
![Page 22: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/22.jpg)
Isomorphic abelian groups
![Page 23: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/23.jpg)
The general zero-sum problem
![Page 24: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/24.jpg)
![Page 25: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/25.jpg)
![Page 26: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/26.jpg)
![Page 27: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/27.jpg)
![Page 28: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/28.jpg)
![Page 29: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/29.jpg)
![Page 30: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/30.jpg)
A variation
![Page 31: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/31.jpg)
A variation
![Page 32: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/32.jpg)
A new challenge
![Page 33: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/33.jpg)
Calculate Reset
![Page 34: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/34.jpg)
A simple lower bound
![Page 35: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/35.jpg)
Erdös,Ginzburg,Ziv Theorem
![Page 36: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/36.jpg)
Generalizing EGV
![Page 37: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/37.jpg)
Generalizing EGV
![Page 38: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/38.jpg)
![Page 39: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/39.jpg)
![Page 40: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/40.jpg)
Other variations
![Page 41: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/41.jpg)
An application of zero-sum problems Zero-sum problems were used to
show that there are infinitely many Carmichael numbers
![Page 42: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/42.jpg)
Fermat’s “little” theorem
![Page 43: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/43.jpg)
Pseudoprimes
![Page 44: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/44.jpg)
Carmichael numbers
![Page 45: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/45.jpg)
Carmichael numbers
![Page 46: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/46.jpg)
Infinitely many Carmichael numbers
![Page 47: On zero-sum problems](https://reader035.fdocuments.us/reader035/viewer/2022062222/5681675e550346895ddc2e6c/html5/thumbnails/47.jpg)
References
Any questions?