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A note on p-adic valuations of Schenker sums

Piotr Miska — 2015

Colloquium Mathematicae

A prime number p is called a Schenker prime if there exists n ∈ ℕ₊ such that p∤n and p|aₙ, where a = 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 v ( 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...

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