Problem
NT-B2-M08-P017 Fourth Powers Modulo \(5\)
#17
★★★★☆ Level 4 of 5
Prove that \(x^4+y^4=5z^4\) has no nonzero integer solutions.
Find the fourth powers modulo \(5\).
A fourth power modulo \(5\) is \(0\) or \(1\). Therefore \(x^4+y^4\equiv 0\pmod 5\) implies \(5\mid x\) and \(5\mid y\). Then \(5^4\mid x^4+y^4=5z^4\), so \(5^3\mid z^4\), hence \(5\mid z\). Dividing all variables by \(5\), we get a smaller nonzero solution of the same equation. Contradiction.
A companion to the previous problem, but with another prime and a shorter residue table.