A note on $p$-adic valuations of Schenker sums

Piotr Miska¹

¹ Institute of Mathematics Faculty of Mathematics and Computer Science Jagiellonian University in Cracow Łojasiewicza 6 30-348 Kraków, Poland

Colloquium Mathematicum, Tome 140 (2015) no. 1, pp. 5-13.

Résumé

A prime number $p$ is called a Schenker prime if there exists $n\in\mathbb{N}_+$ such that $p\nmid n$ and $p\,|\, a_n$, where $a_n = \sum_{j=0}^{n}(n!/j!)n^j$ is a so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning $p$-adic valuations of $a_n$ when $p$ is a Schenker prime. In particular, they conjectured that for each $k\in\mathbb{N}_+$ there exists a unique positive integer $n_k5^k$ such that $v_5(a_{m\cdot 5^k + n_k})\geq k$ for each nonnegative integer $m$. We prove that for every $k\in\mathbb{N}_+$ the inequality $v_5(a_n)\geq k$ has exactly one solution modulo $5^k$. This confirms the above conjecture. Moreover, we show that if $37\nmid n$ then $v_{37}(a_n)\leq 1$, which disproves the other conjecture of the above mentioned authors.

Détail

DOI : 10.4064/cm140-1-2

Mots-clés : prime number called schenker prime there exists mathbb nmid where sum so called schenker sum amdeberhan callan moll formulated conjectures concerning p adic valuations schenker prime particular conjectured each mathbb there exists unique positive integer cdot geq each nonnegative integer prove every mathbb inequality geq has exactly solution modulo nbsp confirms above conjecture moreover nmid leq which disproves other conjecture above mentioned authors