, Volume 80, Issue 1, pp 373-392

Explicit complete solution in integers of a class of equations (ax 2 -b )(ay 2 -b) =z 2 -c

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abstract

In this paper we will study the equation for arbitrary integersa ≠ 0,c andb = ±1, ±2 or ±4 Whenb = ±4, we suppose c is divisible by 4. The paper will provide one with a method for finding algorithmically all integral non-trivial solutions of the title equations, where an explicit unit of $Q(\sqrt {a^2 n^2 - ab} )$ plays an important role.

Dedicated to Dr. Taro Morishima