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

Authors: Kevin G. Hare, Shanta Laishram and Thomas Stoll
Journal: Proc. Amer. Math. Soc. 139 (2011), 39-49
MSC (2010): Primary 11B99, 11Y55
Published electronically: August 19, 2010
MathSciNet review: 2729069
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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 $ \displaystyle{ \liminf_{n\to\infty}} \frac{s_2(n^2)}{s_2(n)} = 0$. He conjectured that, just 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} + \dots + a_0 \in \mathbb{Z}[x]$ with $ h\geq 2$ and $ a_h>0$ and any base $ q$,

$\displaystyle \liminf_{n\to\infty} \frac{s_q(p(n))}{s_q(n)}=0.$

For any $ \varepsilon > 0$ we give a bound on the minimal $ n$ such that the ratio $ s_q(p(n))/ s_q(n) < \varepsilon$. Further, we give lower bounds for the number of $ n < N$ such that $ s_q(p(n))/s_q(n) < \varepsilon$.

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Additional Information

Kevin G. Hare
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

Shanta Laishram
Affiliation: Department of Mathematics, Indian Institute of Science Education and Research, Bhopal, 462 023, India

Thomas Stoll
Affiliation: Institut de Mathématiques de Luminy, Université de la Méditerranée, 13288 Marseille Cedex 9, France

Received by editor(s): January 22, 2010
Published electronically: August 19, 2010
Additional Notes: The first author was partially supported by NSERC; computational support was provided by a CFI/OIT grant
The second author was partially supported by an APART grant of the Austrian Academy of Sciences
Communicated by: Ken Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.