Vol. 46, No. 1, 1973

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ISSN: 0030-8730
The diophantine equation x2 + D = pn

Ronald Alter and K. K. Kubota

Vol. 46 (1973), No. 1, 11–16
Abstract

Let D 3( mod 4) be a positive square free integer greater than 3 which is not a multiple of the odd prime p. If d is the order of a prime ideal divisor of (p) in the class group of the quadratic field Q(√ −-D-), then in order for the diophantine equation x2 + D = pn to have a solution in integers, it is necessary and sufficient that (D∕p) = 1 and that either (i) 4pd D be a square and 3 pd D = ±2 or (ii) pd D be a square. Conditions which guarantee the uniqueness of the solution are given. A linear recurrence is used in the proof.

Mathematical Subject Classification
Primary: 10B15
Milestones
Received: 18 January 1972
Published: 1 May 1973
Authors
Ronald Alter
K. K. Kubota