Original scientific paper

Products of Fermat or Mersenne numbers in some sequences

Mohamadou Bachabi ; Institut de Mathematiques et de Sciences Physiques (IMSP), Universite d’Abomey-Calavi (UAC), Dangbo, Benin
Alain Togbé ; Department of Mathematics and Statistics, Purdue University Northwest, Hammond, USA

Abstract

Let \(P_{n}\) 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
\(P_{n}=(2^{a}-1)(2^{b}-1)\),
\(E_{n}=(2^{a}-1)(2^{b}-1\),
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

Hrčak ID:

321269

URI

https://hrcak.srce.hr/321269

Publication date:

7.10.2024.

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