For any given positive integer $n$, prove $(n^2 +n +1)$ cannot be a perfect square.
Solve in integers $y^2=x^4 + x^3 + x^2 +x +1$.
Find all the ordered integers $(a, b, c)$ which satisfy $a+b+c=450$ and $\sqrt{a+\sqrt{b}}+\sqrt{a-\sqrt{b}}=2c$.
Solve in integers $x^3 + (x+1)^3 + \cdots + (x+7)^3 = y ^3$
Solve this equation in positive integers $$x^3 - y^3 = xy + 61$$