The Diophantine equation x² = 4qⁿ−4q +1

Christopher Skinner

Vol. 139 (1989), No. 2, 303–309

DOI: 10.2140/pjm.1989.139.303

Abstract

In this paper all integral solutions to the equation x² = 4qⁿ − 4q + 1 when q is an odd prime are determined. This is done by working in a quadratic field, using the unique factorization of ideals to reduce the problem to one about certain binary linear recurrences. One of the results is that the equation has no solutions with n > 2 if q > 5.

Mathematical Subject Classification 2000

Primary: 11D41

Milestones

Received: 22 April 1988

Revised: 27 June 1988

Published: 1 October 1989