On the solutions of a family of quartic Thue equations

Togbé, Alain

doi:10.1090/S0025-5718-99-01100-X

On the solutions of a family of quartic Thue equations
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by Alain Togbé;

Math. Comp. 69 (2000), 839-849

DOI: https://doi.org/10.1090/S0025-5718-99-01100-X

Published electronically: May 17, 1999

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Abstract:

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 $n\leq 5\times 10^6$ and $n \geq N=1.191\times 10^{19}$, with $n, n+2, n^2+4$ square-free, the Thue equation \[ \Phi _n(x,y)=x^4 - n^2 x^3 y -(n^3+2n^2+4n+2) x^2 y^2 - n^2 x y^3 + y^4 = 1 \] has no integral solution except the trivial ones: $(1,0),\; (-1,0),\; (0,1),\; (0,-1)$.

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