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dd> Supawadee Prugsapitak, Nattaporn Thongngam,
On the Diophantine equations of the form x^2-kxy+y^2+kx=0 and x^2-kxy+y^2+2kx=0.
Int. J. Math. Comput. Sci., 20, no. 2, (2025), 685-687

DOI:

https://doi.org/10.69793/ijmcs/02.2025/prugsapitak

Keywords and phrases:

Diophantine Equations, Quadratic Diophantine Equations, Infinitely Many Solutions.

Abstract:

For an integer k, define S(k) as the set of integers l where the equation x^2-lxy+y^2+kx=0 has infinitely many positive integer solutions. We demonstrate that for positive integer k, k \in S(k) if and only if k=3, 4, 5; and for k \equiv 0,1,3 mod(4), k \in S(2k) if and only if k=3, 4, 5.