<s>[INST] Infinite positive integer solutions of the equation: $x^2+x+1=(y^2+y+1)(n^2+n+1)$ Could anybody help me some ideas on the below problem:
Let $n$ be a positive integer. Prove that there are infinite pairs of positive integer $(x\, y)$ such that
$$x^2+x+1=(y^2+y+1)(n^2+n+1).$$
Thanks in advance. 
 [/INST] To solve the Diophantine equation.
$$x^2+x+1=(z^2+z+1)(y^2+y+1)$$
It is necessary to use the solutions of the Pell equation. $p^2-(z^2+z+1)s^2=\pm1$
Then the solutions can be written as follows.
$$x=\mp((z+1)p^2+(z^2+z+1)(zs-p)s)$$
$$y=\mp((2z+1)p-(z^2+z+1)s)s$$
For positive you need to take decisions at $-1$.
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