Integral representations for Mersenne and Horadam-Fermat numbers

Year 2024, Volume: 9 Issue: 3, 185 - 200, 31.12.2024

Ahmet İpek

https://doi.org/10.30931/jetas.1553048

Abstract

In this note we first derive integral representations for Mersenne numbers $M_{kn}$ and Horadam-Fermat numbers $\mathcal{F}_{kn}$, then we use those to provide integral representations for Mersenne numbers $M_{kn+r}$ and Horadam-Fermat numbers $\mathcal{F}_{kn+r}$, where $n\in\mathbb{Z}_{>0}=\{1,2,3,\ldots\}$ is a non-negative integer, $k\in\mathbb{Z}_{>0}=$ $\{1,2,3,\ldots\}$ is an arbitrary but fixed positive integer, while $r\in\mathbb{Z}_{\geqslant0}$ is an arbitrary but fixed non-negative integer.

Keywords

Mersenne number, Fermat number, Horadam number, Integral representations.

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