Problem
NT-B2-M08-P016 Fourth Powers Modulo \(3\)
#16
★★★★☆ Level 4 of 5
Prove that \(x^4+y^4=3z^4\) has no nonzero integer solutions.
A fourth power modulo \(3\) is \(0\) or \(1\).
From \(x^4+y^4\equiv 0\pmod 3\), both \(x\) and \(y\) are divisible by \(3\). Then the left side is divisible by \(81\), so \(3z^4\) is divisible by \(81\), and \(z\) is divisible by \(3\). Dividing \(x,y,z\) by \(3\) preserves the equation and gives a smaller nonzero solution. Contradiction.
Shows why homogeneity matters in descent.