A note on the Diophantine equation \(x^2+q^m=c^n\)

Authors

  • N. Terai Ashikaga Institute of Technology

Keywords:

Diophantine equation, integer solution

Abstract

Let \(q\) be an odd prime such that \(q^t+1=2c^s\), where \(c,t\) are positive integers and \(s=1,2\). We show that the Diophantine equation \(x^2+q^m=c^n\) 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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