Abstract
For any fixed positive integer D which is not a square, let (u, υ) = (u 1, υ 1) be the fundamental solution of the Pell equation u 2 − Dυ 2 = 1. Further let \(\mathbb{D}\) be the set of all positive integers D such that D is odd, D is not a square and gcd(D, υ 1) > max(1, √D/8). In this paper we prove that if (x, y, z) is a positive integer solution of the equation x y + y x = z 2 satisfying gcd(x, y) = 1 and xy is odd, then either \(x \in \mathbb{D}\) or \(y \in \mathbb{D}\).
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Communicated by Attila Pethő
This work is supported by the N. S. F. (2009JM1006) of Shaanxi Province, P.R. China.
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Xiaoying, W. The exponential diophantine equation x y + y x = y 2 with xy odd. Period Math Hung 66, 193–200 (2013). https://doi.org/10.1007/s10998-013-5083-5
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DOI: https://doi.org/10.1007/s10998-013-5083-5