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In this paper, we show that if p and q are positive integers, then the polynomial exponential equation p x + q x = y 2 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's method for finding rational points on curves of higher genus. 2010 Mathematics Subject Classification. 11D61, 11D41.doi:10.3336/gm.50.2.03 fatcat:4r6r7axmxjdgxmcchvpcunmbba