A parametric family of quintic Thue equations

Abstract:

For an integral parameter $t \in \mathbb {Z}$ we investigate the family of Thue equations \begin{multline*} F(x,y) = x^{5} + (t-1)^{2}x^{4}y - (2t^{3}+4t+4)x^{3}y^{2}\ + (t^{4}+t^{3}+2t^{2}+4t-3)x^{2}y^{3} + (t^{3}+t^{2}+5t+3)xy^{4} + y^{5} = \pm 1 , \end{multline*} originating from Emma Lehmer’s family of quintic fields, and show that for $|t| \ge 3.28 \cdot 10^{15}$ the only solutions are the trivial ones with $x=0$ or $y=0$. Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest.