The Exponential Diophantine Equation 4 m 2 + 1 x + 5 m 2 - 1 y = ( 3 m )‎ z

Abstract EN

Let m be a positive integer.

In this paper, using some properties of exponential diophantine equations and some results on the existence of primitive divisors of Lucas numbers, we prove that if m > 90 and 3 | m , then the equation 4 m 2 + 1 x + 5 m 2 - 1 y = ( 3 m ) z has only the positive integer solution ( x , y , z ) = ( 1 , 1 , 2 ) .