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A note on pp-adic valuations of the Schenker sums

Abstract

A prime number pp is called a Schenker prime if there exists such n∈N+n\in\mathbb{N}_+ that p∀np\nmid n and p∣anp\mid a_n, where an=βˆ‘j=0nn!j!nja_n = \sum_{j=0}^{n}\frac{n!}{j!}n^j is so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning pp-adic valuations of ana_n in case when pp is a Schenker prime. In particular, they asked whether for each k∈N+k\in\mathbb{N}_+ there exists the unique positive integer nk<pkn_k<p^k such that vp(amβ‹…5k+nk)β‰₯kv_p(a_{m\cdot 5^k + n_k})\geq k for each nonnegative integer mm. We prove that for every k∈N+k\in\mathbb{N}_+ the inequality v5(an)β‰₯kv_5(a_n)\geq k has exactly one solution modulo 5k5^k. This confirms the first conjecture stated by the mentioned authors. Moreover, we show that if 37∀n37\nmid n then v37(an)≀1v_{37}(a_n)\leq 1, what means that the second conjecture stated by the mentioned authors is not true

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