Problem
NT-B1-M03-P018 Fourth Powers Modulo 16
#18
★★★☆☆ Level 3 of 5
Prove that \(x^4+y^4=16z+15\) has no integer solutions.
A fourth power modulo \(16\) is \(0\) or \(1\).
If \(x\) is even, then \(x^4\equiv0\pmod{16}\); if \(x\) is odd, then \(x^2\equiv1\) or \(9\pmod{16}\), and \(x^4\equiv1\pmod{16}\). Thus \(x^4+y^4\) can have residue \(0,1,2\), but not \(15\). The right side has residue \(15\). Contradiction.
Introduces the useful modulus \(16\).