Products of Fermat or Mersenne numbers in some sequences

Mohamadou Bachabi; Alain Togbé

Vol. 29 No. 2 (2024)

Regular Articles

Products of Fermat or Mersenne numbers in some sequences

Published 2024-10-04

Mohamadou Bachabi
Alain Togbé

Mohamadou Bachabi
Institut de Mathematiques et de Siences Physiques

Alain Togbé
Department of Mathematics and Statistics, Purdue University Northwest, 2200 169th Street, Hammond, IN 46323, USA

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Abstract

Let $\mathcal{P}_n$ be an $n$-th Padovan number, $E_n$ an $n$-th Perrin number, and $N_n$ an $n$-th Nayarana number. In this paper, we solve the Diophantine equations $$ \mathcal{P}_n=(2^a-1)(2^b-1), $$ $$ E_n=(2^a-1)(2^b-1), $$and $$ N_n=(2^a\pm 1)(2^b\pm 1), $$ in positive unknowns $n, a$, and $b$. Therefore, we determine the Padovan or Perrin numbers that are products of two Mersenne numbers and the Nayarana numbers that are Mersenne numbers and two Fermat numbers.

Keywords

Diophantine equations, Padovan sequence, Perrin sequence, Narayana sequence, linear forms in logarithms, reduction method

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Supplementary File(s)

4960-14944-1-SP - final