Find all integer solutions of $1+x+x^2+x^3=y^2$ I need some help on solving this problem:
Find all integer solutions for this following equation:
$1+x+x^2+x^3=y^2$
My attempt:
Clearly $y^2 = (1+x)(1+x^2)$, assuming the GCD[$(1+x), (1+x^2)] = d$, then if $d>1$, $d$ has to be power of 2. This implies that I can assume: $1+x=2^s*a^2, 1+x^2=2^t*b^2$. If $t=0$ then it is easy to finish. Considering $t>0$, we can get $t=1$ (simple steps only), so I come up with a "Pell-related" equation .. Then I get sticking there. It has a solution $x=7$, so I guess it's not easy to find the rest.
Please help.

This problem, and its “inverse” (q.v. Finding all solutions to $y^3 = x^2 + x + 1$ with $x,y$ integers larger than $1$) were posed by Fermat in 1657 (see, for example, Mahoney pg. 337). Both lead to Pell equations (q.v. Solutions to $p+1=2n^2$ and $p^2+1=2m^2$ in Natural numbers.), which was exactly what Fermat was trying to get his contemporaries to study with him.
