SQUARES FROM SUMS OF FIXED POWERS

In this paper, we show that if p and q are positive integers, then the polynomial exponential equation px+qx=y2 can have at most two solutions in positive integer x and y. If such solutions exists, we are able to precisely characterize them. Our proof relies upon a result of Darmon and Merel, and Chabauty\u27s method for finding rational points on curves of higher genus