Forbidden Integer Ratios of Consecutive Power Sums

Ioulia N. Baoulina () and Pieter Moree ()
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Ioulia N. Baoulina: Moscow State Pedagogical University, Department of Mathematics
Pieter Moree: Max-Planck-Institut für Mathematik

A chapter in From Arithmetic to Zeta-Functions, 2016, pp 1-30 from Springer

Abstract: Abstract Let S k (m): = 1 k + 2 k + ⋯ + (m − 1) k denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for m ≥ 4 the ratio S k (m + 1)∕S k (m) of two consecutive power sums is never an integer. We will develop some techniques that allow one to exclude many integers ρ as a ratio and combine them to exclude the integers 3 ≤ ρ ≤ 1501 and, assuming a conjecture on irregular primes to be true, a set of density 1 of ratios ρ. To exclude a ratio ρ one has to show that the Erdős–Moser type equation (ρ − 1)S k (m) = m k has no non-trivial solutions.

Keywords: Consecutive power sums; Erdős–Moser type equation; Primary 11D61; Secondary 11A07 (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-28203-9_1

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DOI: 10.1007/978-3-319-28203-9_1

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