The ternary exponential Diophantine equation concerning Pythagorean triplets

Abstract

Let r be a positive integer with r > 1, and let m be a positive even integer. Further let a = |V (m, r)|, b = |U(m, r)| and c = m² + 1, where V (m, r) + U(m, r) √-1 = (m + √-1)^r. In this paper, using the Gel′ fond-Baker method, we prove that if 2 | r and m > max(10¹⁵, 2r³), then the equation a^x + b^y = c^z has only the positive integer solution (x, y, z) = (2, 2, r).

Keywords: Ternary exponential Diophantine equation, Pythagorean triplet, Gel′ fond- Baker method.