On the solutions of a family of quartic Thue equations

Alain Togbé
1999 Mathematics of Computation  
In this paper, we solve a certain family of diophantine equations associated with a family of cyclic quartic number fields. In fact, we prove that for n ≤ 5 × 10 6 and n ≥ N = 1.191 × 10 19 , with n, n + 2, n 2 + 4 square-free, the Thue equation Φn(x, y) = x 4 − n 2 x 3 y − (n 3 + 2n 2 + 4n + 2)x 2 y 2 − n 2 xy 3 + y 4 = 1 has no integral solution except the trivial ones:
doi:10.1090/s0025-5718-99-01100-x fatcat:yvk5cx3asfcq3ng3gcjfc42hq4