Abstract
Let S k (m): = 1k + 2k + ⋯ + (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.
To the memory of Prof. Wolfgang Schwarz
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Acknowledgements
This paper was begun during the stay of the first author in February–April 2014 at the Max Planck Institute for Mathematics. She likes to thank for the invitation and the pleasant research atmosphere. The second author was introduced to the subject around 1990 by the late Jerzy Urbanowicz. He will never forget his interest, help and kindness. Further he thanks Prof. T.N. Shorey for helpful discussions in the summer of 2014. We would like to heartily thank Bernd Kellner for helpful e-mail correspondence and patiently answering our Bernoulli number questions. A. Ciolan, P. Tegelaar and W. Zudilin kindly commented on an earlier version of this paper.The cooperation of the authors has its origin in them having met, in 2012, at ELAZ in Schloss Schney (which was also attended by Prof. W. Schwarz). This contribution is our tribute to Prof. W. Schwarz, who was one of the initiators of the ELAZ conference series.
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Baoulina, I.N., Moree, P. (2016). Forbidden Integer Ratios of Consecutive Power Sums. In: Sander, J., Steuding, J., Steuding, R. (eds) From Arithmetic to Zeta-Functions. Springer, Cham. https://doi.org/10.1007/978-3-319-28203-9_1
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