Practical solution of the Diophantine equation 𝑦²=𝑥(𝑥+2^{𝑎}𝑝^{𝑏})(𝑥-2^{𝑎}𝑝^{𝑏})

Draziotis, Konstantinos; Poulakis, Dimitrios

doi:10.1090/S0025-5718-06-01841-2

Practical solution of the Diophantine equation $y^2 = x(x+2^ap^b)(x-2^ap^b)$
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by Konstantinos Draziotis and Dimitrios Poulakis PDF

Math. Comp. 75 (2006), 1585-1593 Request permission

Abstract:

Let $p$ be an odd prime and $a$, $b$ positive integers. In this note we prove that the problem of the determination of the integer solutions to the equation $y^2 = x(x+2^ap^b)(x-2^ap^b)$ can be easily reduced to the resolution of the unit equation $u+\sqrt {2}v = 1$ over $\mathbb {Q}(\sqrt {2},\sqrt {p})$. The solutions of the latter equation are given by Wildanger’s algorithm.

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