Number Theory Seminar

Tuesday, February 6, 2018 1:30 pm - 1:30 pm EST

Jeffrey Shallit, David R. Cheriton School of Computer Science, University of Waterloo

"Waring's theorem for binary k'th powers and palindromes"

Recently there has been some interest in analogues of Waring's theorem for other kinds of sets analogous to powers of integers. For example, Banks proved that every natural number is the sum of at most 49 numbers whose base-10 expansion is a palindrome, and this was improved to 3 for all bases b ≥ 5 by Cilleruelo, Luca, and Baster.

In this talk I will discuss some recent results about palindromes in bases 2, 3, 4 and binary k'th powers. (A binary k'th power is an integer whose base-2 expansion consists of k consecutive  identical blocks.) In particular, I'll show that for all k ≥ 1 there exists a constant W(k) such that every sufficiently large multiple of gcd(k,2k - 1)$ is the sum of W(k) binary k'th powers.

This is joint work with Aayush Rajasekaran, Dirk Nowotka, Parthasarathy Madhusudan, Tim Smith, Daniel Kane, and Carlo Sanna.

MC 5417