A note on p-adic valuations of Schenker sums

• Autorzy: Piotr Miska
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11B83: Special sequences and polynomials
• 11B50: Sequences (mod m )

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2015
• Tom: 140
• Numer: 1
• Strony: 5-13

Daty

wydano

2015

Twórcy

autor

Piotr Miska

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

Identyfikatory

DOI

10.4064/cm140-1-2