Student Number Theory seminar

J.C. Saunders, Department of Pure Math, University of Waterloo

"Sums of Digits in q-ary expansions Part 2"

Let sq(n) denote the sum of the digits of a number n in base q. For example, s2(n) represents the number of 1s in the binary expansion of n. In 1978, Kenneth B. Stolarsky showed that lim inf n!1 s2(n2) s2(n)
= 0 using bounds obtained from analytical methods. In the last presentation we showed that the ratio s2(n2) s2(n) can indeed hit every positive rational number. In this presentation, we show that the same is
true for this ratio in general base q, that is sq(n2) sq(n) : n 2 N
= Q+: for all q 2. If time permits, we will also look at powers with certain fractional exponents to derive similar conclusions.