Forbidden integer ratios of consecutive power sums

Abstract

Let $S_k(m):=1^k+2^k+\cdots +(m-1{)}^k$ denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for $m\ge 4$ 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 $\rho$ as a ratio and combine them to exclude the integers $3\le \rho \le 1501$ and, assuming a conjecture on irregular primes to be true, a set of density $1$ of ratios $\rho$. To exclude a ratio $\rho$ one has to show that the Erdős-Moser type equation $(\rho -1)S_k(m)=m^k$ has no non-trivial solutions.

28 pages, 3 tables; accepted for publication in the book "From Arithmetic to Zeta-Functions - Number Theory in Memory of Wolfgang Schwarz"

References in corpus (5)

• On irregular prime power divisors of the Bernoulli numbers
• The Erdős--Moser equation $1^k+2^k+...+(m-1{)}^k=m^k$ revisited using continued fractions
• On stronger conjectures that imply the Erdös-Moser conjecture
• Moser's mathemagical work on the equation 1^k+2^k+...+(m-1)^k=m^k
• Forbidden integer ratios of consecutive power sums

Cited by in corpus (2)

• Forbidden integer ratios of consecutive power sums
• Integer ratios of consecutive alternating power sums