Research on the Number of Solutions to a Special Type of Diophantine Equation ( a x −1)( b y −1) = 2 z 2

Shu-Hui Yang ()
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Shu-Hui Yang: Fundamental Teaching Department of Computer and Mathematics, Shenyang Normal University, Shenyang 110034, China

Mathematics, 2023, vol. 11, issue 11, 1-8

Abstract: Let b be an odd number. By using elementary methods, we prove that: (1) When x is an odd number and y is an even number, the Diophantine equation ( 2 x − 1 ) ( b y − 1 ) = 2 z 2 has no positive integer solution except when b is two special types of odd number. (2) When x is an odd number and b ≡ ± 3 ( mod 8 ) , the Diophantine equation ( 2 x − 1 ) ( b y − 1 ) = 2 z 2 has no positive integer solution except where b = 3 and is another special type of the odd number.

Keywords: recursive sequence method; quadratic residual method; Diophantine equation (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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