On the exponential Diophantine equation $(a^n-1)(b^n-1)=x^2$

Abstract

Year: 2010 Vol.: 77 Fasc.: 3-4 Title: On the exponential diophantine equation (an i 1)(bn i 1) = x2 Author(s): Li Lan and L¶aszl¶o Szalay Let a and b be ¯xed positive integers such that a 6= b and min(a; b) > 1. In this paper, we combine some divisibility properties of the solutions of Pell equations with elementary arguments to prove that if a ´ 2 (mod 6) and b ´ 0 (mod 3), then the title equation (an i1)(bn i1) = x2 has no positive integer solution (n; x). Moreover, we show that in case of a ´ 2 (mod 20) and b ´ 5 (mod 20), where b i 1 is a full square, the only possible solution belongs to n = 1. Address: Li Lan Department of Mathematics Xi'an University of Arts & Science Xi'an 710065 P.R. China Address: L¶aszl¶o Szalay Institute of Mathematics and Statistics University of West Hungary Sopron Hungary