Generalizations of Euler’s Theorem to Matrices

Start: 04/15/2015 - 4:15pm
End  : 04/15/2015 - 5:15pm


Bogdan Petrenko, Eastern Illinois University


This talk will be based on my work with Marcin Mazur (Binghamton University). V.I. Arnold’s investigations of discrete dynamical systems led him to many intriguing number-theoretical ques- tions. In particular, he was motivated by dynamical considerations when he asked whether it would be possible to extend Euler’s totient theorem to square matrices with integer entries. This theorem states that if a and n ≥ 2 are relatively prime positive integers, then n divides aφ(n) − 1, where φ(n) is the number of integers from 1,2,...,n−1 that are relatively prime to n. When n is a power of a prime number, such an extension was obtained by W. J ̈anichen in 1921, and it was independently rediscovered by I. Schur in 1937, and by V.I. Arnold in 2001. In this talk, I will present such an extension for all 2-by-2 integer matrices and all n ≥ 2; this extension implies Euler’s totient theorem. We will also see that such an extension does not exist for all r-by-r integer matrices, where r ≥ 3 is fixed, and all n ≥ 2. It is very likely, however, that for each such r, there exist interesting two-term congruences that remain to be discovered.

Shanahan Center for Teaching and Learning (SCTL), at Harvey Mudd, Basement, B460

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