• A note on the exponential diophantine equation $(a^{n} − 1)(b^{n} − 1) = x^{2}$

• # Fulltext

https://www.ias.ac.in/article/fulltext/pmsc/129/05/0069

• # Keywords

Pell equation; exponential diophantine equation; Lucas sequence

• # Abstract

In 2002, Luca and Walsh (J. Number Theory 96 (2002) 152–173) solved the diophantine equation for all pairs $(a, b)$ such that $2\leq a$ < $b\leq 100$ with some exceptions. There are sixty nine exceptions. In this paper, we give some new results concerning the equation $(a^{n}−1)(b^{n}−1) = x^{2}$. It is also proved that this equation has no solutions if $a, b$ have opposite parity and $n$ >$4$ with $2|n$. Here, the equation is also solved for the pairs $(a, b) = (2, 50), (4, 49), (12, 45), (13, 76), (20, 77), (28, 49), (45, 100)$. Lastly, we show that when b is even, the equation $(a^{n} − 1)(b^{2n}a^{n} − 1) = x^{2}$ has no solutions $n, x$.

• # Author Affiliations

1. Department of Mathematics, Faculty of Arts and Science, Sakarya University, Sakarya, Turkey

• # Proceedings – Mathematical Sciences

Current Issue
Volume 129 | Issue 5
November 2019

• # Editorial Note on Continuous Article Publication

Posted on July 25, 2019