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Alanod M. Sibih,
Note on the Diophantine Equation x^2 + 4pxy - p^2(q^2 - 4)y^2 = k^t.
Int. J. Math. Comput. Sci., 20, no. 3, (2025), 751-753

DOI:

https://doi.org/10.69793/ijmcs/03.2025/alanod

Keywords and phrases:

Diophantine equations, Pell-type equations, continued fractions, recurrence relations, quadratic fields.

Abstract:

In this paper, we investigate positive integer solutions to the Diophantine equation: x^2 + 4pxy - p^2(q^2 - 4)y^2 = k^t, where p, q, k, t are positive integers. Using the substitution z = x + 2py, we transform the equation into a Pell-type form z^2 - p^2q^2y^2 = k^t. We then derive explicit formulas for fundamental solutions and establish recurrence relations to generate all positive integer solutions. The results generalize previous work on Pell-type equations with mixed terms.