Stolarsky's conjecture and the sum of digits of polynomial values

2010/01/23 by K. G. Hare, Hare, K. G., S. Laishram +3
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1001.4169

13 pages

arxiv created 2010/01/23 · arxiv updated 2010/02/26

Abstract

Let s_q(n) denote the sum of the digits in the q-ary expansion of an integer n. In 1978, Stolarsky showed that \liminf_n→∞ (s₂(n²))/(s₂(n)) = 0. He conjectured that, as for n², this limit infimum should be 0 for higher powers of n. We prove and generalize this conjecture showing that for any polynomial p(x)=a_h x^h+a_h-1 x^h-1 + ... + a₀ ∈ \Z[x] with h≥ 2 and a_h>0 and any base q, \liminf_n→∞ (s_q(p(n)))/(s_q(n))=0. For any ε> 0 we give a bound on the minimal n such that the ratio s_q(p(n))/s_q(n) < ε. Further, we give lower bounds for the number of n < N such that s_q(p(n))/s_q(n) < ε.

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