The complete solution of the Diophantine equation $$(4m^2+1)^x+(5m^2-1)^y=(3m)^z$$ ( 4 m 2 + 1 ) x + ( 5 m 2 - 1 ) y = ( 3 m ) z

Abstract

In this short note we give all solutions to the exponential Diophantine equation \((4m^2+1)^x+(5m^2-1)^y=(3m)^z\) where \(20<m<90\) and \(m\equiv 3\)\((mod\quad 6)\). In view of earlier theorems of Terai, and of Su and Li, this result yields a complete solution set to a problem of Terai.