Complete solution of the diophantine equation $X^2+1=d{Y}^4$ and a related family of quartic Thue equations

Abstract

In this paper, we use the method of Thue and Siegel, based on explicit Pade approximations to algebraic functions, to completely solve a family of quartic Thue equations. From this result, we can also solve the diophantine equation in the title. We prove that this equation has at most one solution in positive integers when $d\ge 3$. Moreover, when such a solution exists, it is of the form $(u,\sqrt{v})$ where $(u,v)$ is the fundamental solution of $X^2+1=d{Y}^2$.

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