Abstract
Let p be an odd prime. By using the elementary methods we prove that: (1) if 2 ∤ x, p = ±3 (mod 8), the Diophantine equation (2x − 1)(py − 1) = 2z2 has no positive integer solution except when p = 3 or p is of the form \(p = 2a_0^2 + 1\), where a0 > 1 is an odd positive integer. (2) if 2 ∤ x, 2 ∣; y, y ≠ 2, 4, then the Diophantine equation (2x − 1)(py − 1) = 2z2 has no positive integer solution.
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The author would like to thank the anonymous reviewers for their valuable suggestions.
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The research has been supported by Guizhou Provincial Science and Technology Foundation (Grant No. QIANKEHEJICHU[2019]1221) and the National Natural Science Foundation of China (Grant No. 11261060)
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Tong, R. On the Diophantine equation (2x − 1)(py − 1) = 2z2. Czech Math J 71, 689–696 (2021). https://doi.org/10.21136/CMJ.2021.0057-20
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DOI: https://doi.org/10.21136/CMJ.2021.0057-20