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Article Dans Une Revue Proceedings of the American Mathematical Society Année : 2011

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

Sur une conjecture de Stolarsky et la somme des chiffres des valeurs polynomiales

Résumé

Let s_q (n) denote the sum of the digits in the q-ary expansion of an integer n. In 1978, Stolarsky showed that lim inf n→∞ 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 ∈ Z[x] with h ≥ 2 and a h > 0 and any base q, lim inf 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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Dates et versions

hal-01278726 , version 1 (24-02-2016)

Identifiants

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Kevin G. Hare, Shanta Laishram, Thomas Stoll. Stolarsky’s conjecture and the sum of digits of polynomial values. Proceedings of the American Mathematical Society, 2011, 139 (1), pp.39-49. ⟨10.1090/S0002-9939-2010-10591-9⟩. ⟨hal-01278726⟩
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