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A modular approach to the generalized ramanujan-nagell equation

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2022-08-20

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Elsevier

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Abstract

Let k be a positive integer. In this paper, using the modular approach, we prove that if k & EQUIV; 0 (mod 4), 30 < k < 724 and 2k -1 is an odd prime power, then under the GRH, the equation x2 + (2k -1)y = kz has only one positive integer solution (x, y, z) = (k - 1, 1, 2). The above results solve some difficult cases of Terai's conjecture concerning this equation.(c) 2022 Royal Dutch Mathematical Society (KWG).

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Diophantine equations, Points, Polynomial-exponential diophantine equation, Elliptic curve, S-integral point, Modular approach, Science & technology, Physical sciences, Mathematics

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