On the number of solutions of the generalized Ramanujan-Nagell equation

Bugeaud, Y. ; Shorey, T. N. (2001) On the number of solutions of the generalized Ramanujan-Nagell equation Journal fur die Reine und Angewandte Mathematik (Crelle's Journal), 2001 (539). pp. 55-74. ISSN 0075-4102

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Official URL: http://www.reference-global.com/doi/abs/10.1515/cr...

Related URL: http://dx.doi.org/10.1515/crll.2001.079, 20/09/2001

Abstract

Let D1 and D2 be coprime positive integers and let k be an odd positive integer coprime with D1D2. We consider the Diophantine equation D1x2 + D2 = kn in the unknowns x≥1, n≥1. We give a necessary and sufficient condition on D1, D2 and k under which this equation has at most 2ω(k)−1 solutions where ω(k) denoted the number of distinct prime divisors of k. Thus, under a necessary and sufficient conditon, the equation has at most one solution whenever k is a prime. We also consider some related equations and we prove that the Diophantine equation x2+7=4yn has no solution in integers x≥1, y > 2 and n >1.

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