Integral representations for Mersenne and Horadam-Fermat numbers

Authors : Ahmet İpek

Pages : 185-200

Doi:10.30931/jetas.1553048

View : 84 | Download : 119

Publication Date : 2024-12-31

Article Type : Research Paper

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.