Notes on Number Theory and Discrete Mathematics Print ISSN 1310–5132, Online ISSN 2367–8275 Vol. 27, 2021, No. 1, 161–170 DOI: 10.7546/nntdm.2021.27.1.161-170 A note on Mersenne Padovan and Perrin numbers Bir Kafle 1 , Salah Eddine Rihane 2 and Alain Togb´ e 3 1 Department of Mathematics, Statistics, and Computer Science Purdue University Northwest 1401 S, U.S. 421, Westville IN 46391, USA e-mail: bkafle@pnw.edu 2 Department of Mathematics and Computer Science Abdelhafid Boussouf University Mila 43000, Algeria e-mail: salahrihane@hotmail.fr 3 Department of Mathematics, Statistics, and Computer Science Purdue University Northwest 1401 S, U.S. 421, Westville IN 46391, USA e-mail: atogbe@pnw.edu Received: 7 August 2020 Revised: 11 February 2021 Accepted: 1 March 2021 Abstract: In this paper, we determine all the Mersenne numbers which are in the sequences of Padovan and Perrin numbers, respectively. Keywords: Padovan numbers, Perrin numbers, Mersenne numbers, Linear form in logarithms, Reduction method. 2010 Mathematics Subject Classification: 11B39, 11D45, 11J86. 1 Introduction The intersection of sequences was and continue to be an interesting subject of research. In [14], Stein examined the intersection of Fibonacci sequences. In particular, he proved that two Fibonacci sequences generally do not meet and that if they do meet at least three times, then one is simply the tail of the other. In [12] generalized Stein’s work by looking at conditions for fewer than two intersections, exactly two intersections, and more than two intersections. We will continue in the same spirit by studying the intersection between the Mersenne numbers and the Padovan or Perrin numbers. 161