Glasnik Matematicki, Vol. 50, No. 2 (2015), 279-288.

SQUARES FROM SUMS OF FIXED POWERS

Mark Bauer and Michael A. Bennett

Department of Mathematics, University of Calgary, Calgary, AB, Canada
e-mail: mbauer@math.ucalgary.ca

Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada
e-mail: bennett@math.ubc.ca

Abstract.   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's method for finding rational points on curves of higher genus.

2010 Mathematics Subject Classification.   11D61, 11D41.

Key words and phrases.   Diophantine equations, Chabauty's method.

Full text (PDF) (free access)

DOI: 10.3336/gm.50.2.03

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