A note on the Diophantine equation x2+qm=cn

Authors

  • N. Terai Ashikaga Institute of Technology

Keywords:

Diophantine equation, integer solution

Abstract

Let q be an odd prime such that qt+1=2cs, where c,t are positive integers and s=1,2. We show that the Diophantine equation x2+qm=cn has only the positive integer solution (x,m,n)=(c^s−1,t,2s) under some conditions. The proof is based on elementary methods and a result concerning the Diophantine equation (x^n−1)/(x−1)=y^2 due to Ljunggren. We also verify that when 2≤c≤30 with c≠12,24, the Diophantine equation (x^2+(2c−1)^m=c^n has only the positive integer solution (x,m,n)=(c−1,1,2). DOI: 10.1017/S0004972713000981

Published

2014-06-03

Issue

Vol. 90 No. 1 (2014)

Section

Articles
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