A note on the exponential Diophantine equation (aˣ - 1) (bʸ - 1) = az²

Fujita, Yasutsugu and Le, Maohua (2024) A note on the exponential Diophantine equation (aˣ - 1) (bʸ - 1) = az². ANNALES MATHEMATICAE ET INFORMATICAE, 60. pp. 44-53. ISSN 1787-6117

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Abstract

Let a,b be fixed positive integers such that (a mod 8,b mod 8) ∈ {(0, 3),(0,5),(2,3),(2,5),(4,3),(6,5)}. In this paper, using elementary methods with some classical results for Diophantine equations, we prove the following three results: (i) The equation (∗) (ax −1)(by −1) = az2 has no positive integer solutions (x,y,z) with 2 ∤ x and x > 1. (ii) If a = 2 and b ≡ 5 (mod 8), then (∗) has no positive integer solutions (x,y,z) with 2 ∤ x. (iii) If a = 2 and b ≡ 3 (mod 8), then the positive integer solutions (x,y,z) of (∗) with 2 ∤ x are determined. These results improve the recent results of R.-Z. Tong: On the Diophantine equation (2x −1)(py −1) = 2z2, Czech. Math. J. 71 (2021), 689–696. Moreover, under the assumption that a is a square, we prove that (∗) has no positive integer solutions (x,y,z) even with 2 | x in some cases.

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