THE EXPONENTIAL DIOPHANTINE EQUATION n^x + (n + 1)^y = (n + 2)^z REVISITED

Let n be a positive integer. In this paper, we consider the diophantine equation We prove that this equation has only the positive integer solutions ( n , x , y , z ) = (1, t , 1, 1), (1, t , 3, 2), (3, 2, 2, 2). Therefore we extend the work done by Leszczyński ( Wiadom. Mat ., vol. 3, 1959, pp. 37–39) and Makowski ( Wiadom. Mat ., vol. 9, 1967, pp....