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A note on p-adic valuations of Schenker sums

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Miska P.

Colloquium Mathematicum

2015

tom 140

nr 1

5-13

A prime number p is called a Schenker prime if there exists n ∈ ℕ₊ such that p∤n and p|aₙ, where $aₙ = ∑_{j=0}^{n} (n!/j!)n^{j}$ is a so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning p-adic valuations of aₙ when p is a Schenker prime. In particular, they conjectured that for each k ∈ ℕ₊ there exists a unique positive integer $n_{k} < 5^{k}$ such that $v₅(a_{m·5^{k}+n_{k}}) ≥ k$ for each nonnegative integer m. We prove that for every k ∈ ℕ₊ the inequality v₅(aₙ) ≥ k has exactly one solution modulo $5^{k}$. This confirms the above conjecture. Moreover, we show that if 37∤n then $v_{37}(aₙ) ≤ 1$, which disproves the other conjecture of the above mentioned authors.

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1 Colloquium Mathematicum

1 Miska P.

1 2015

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A note on p-adic valuations of Schenker sums