Problem

Source: USAMO 2005, problem 2, Razvan Gelca

Tags: modular arithmetic, algebra proposed, algebra, Diophantine equation, Hi



Prove that the system \begin{align*} x^6+x^3+x^3y+y & = 147^{157} \\ x^3+x^3y+y^2+y+z^9 & = 157^{147} \end{align*} has no solutions in integers $x$, $y$, and $z$.