Show that there exist perfect squares a, b, c such that \(a^2 + b^2 = c^2\).

====== REVISED (2022-04-04) ======

I hope you noticed the day this problem appeared was April fool’s day. Show instead that there do not exist perfect squares a, b, c such that \(a^2 + b^2 = c^2\), provided that a, b, c are nonzero integers.