MMN-1857
A note on some Diophantine equations
Refik Keskin; Merve Güney Duman;Abstract
Let k≥3 be an odd integer. In this paper, we investigate all positive integer solutions of the equations x⁴-kx²y+y²=∓A, x⁴-kx²y+y²=∓A(k²-4), x⁴-(k²-4)y²=∓4A, and x²-(k²-4)y⁴=∓4A with A=∓(k∓2). We show that if k≡1(mod8) and k²-4 be a squarefree integer, then the equation x⁴-kx²y+y²=(k-2)(k²-4) has no positive integer solutions. Moreover, if k²-4 be a squarefree integer, then the equation x⁴-kx²y+y²=-(k+2)(k²-4) has no positive integer solutions.
Vol. 18 (2017), No. 1, pp. 235-249
DOI: 10.18514/MMN.2017.1857