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Forbidden Integer Ratios of Consecutive Power Sums

In: From Arithmetic to Zeta-Functions

Author

Listed:
  • Ioulia N. Baoulina

    (Moscow State Pedagogical University, Department of Mathematics)

  • Pieter Moree

    (Max-Planck-Institut für Mathematik)

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.

Suggested Citation

  • Ioulia N. Baoulina & Pieter Moree, 2016. "Forbidden Integer Ratios of Consecutive Power Sums," Springer Books, in: Jürgen Sander & Jörn Steuding & Rasa Steuding (ed.), From Arithmetic to Zeta-Functions, pages 1-30, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-28203-9_1
    DOI: 10.1007/978-3-319-28203-9_1
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