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

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

2010-01-23

arxiv.org/abs/1001.4169v1

Mathematics

Number Theory

13 pages

Scientific paper

Let $s_q(n)$ denote the sum of the digits in the $q$-ary expansion of an integer $n$. In 1978, Stolarsky showed that $$ \liminf_{n\to\infty} \frac{s_2(n^2)}{s_2(n)} = 0. $$ He conjectured that, as for $n^2$, this limit infimum should be 0 for higher powers of $n$. We prove and generalize this conjecture showing that for any polynomial $p(x)=a_h x^h+a_{h-1} x^{h-1} + ... + a_0 \in \Z[x]$ with $h\geq 2$ and $a_h>0$ and any base $q$, \[ \liminf_{n\to\infty} \frac{s_q(p(n))}{s_q(n)}=0.\] For any $\epsilon > 0$ we give a bound on the minimal $n$ such that the ratio $s_q(p(n))/s_q(n) < \epsilon$. Further, we give lower bounds for the number of $n < N$ such that $s_q(p(n))/s_q(n) < \epsilon$.

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