Rational points on $y^5 = x^4 + x^3 + x^2 + x + 1$

Solution 1:

There are several references on integer (and rational) solutions of the equation $$ y^n=1+x+x^2+\cdots +x^{m-1} $$ for $m>2,\; n>1$, e.g. the article of Li Yu and Maohua Le and the references therein; the article of N. Hirata-Kohno and T.N. Shorey of 1996, "On the equation $(x^m-1)/(x-1)=y^q$ and the references therein, in several articles by Y. Bugeaud, and in many other papers of this kind. I think that if a complete answer is possible, then you will find the answer (and a proof) there in the literature . For $(m,n)=(5,5)$ it is your equation. I believe it is your task now, to go through these many cases which have been treated there.