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 ax+by=cz has only the positive integer solution (x, y, z)=(2, 2, r).