A note on p-adic valuations of Schenker sums

Piotr Miska. "A note on p-adic valuations of Schenker sums." Colloquium Mathematicae 140.1 (2015): 5-13. <http://eudml.org/doc/286349>.

@article{PiotrMiska2015,
abstract = {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.},
author = {Piotr Miska},
journal = {Colloquium Mathematicae},
keywords = {-adic valuation; prime; Schenker sum},
language = {eng},
number = {1},
pages = {5-13},
title = {A note on p-adic valuations of Schenker sums},
url = {http://eudml.org/doc/286349},
volume = {140},
year = {2015},
}

TY - JOUR
AU - Piotr Miska
TI - A note on p-adic valuations of Schenker sums
JO - Colloquium Mathematicae
PY - 2015
VL - 140
IS - 1
SP - 5
EP - 13
AB - 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.
LA - eng
KW - -adic valuation; prime; Schenker sum
UR - http://eudml.org/doc/286349
ER -