Problem
NT-B2-M07-P011 Impossibility Modulo \(4\)
#11
★★★☆☆ Level 3 of 5
Prove that the equation \(x^2+y^2=4xy+3\) has no integer solutions.
Compare both sides modulo \(4\).
An integer square modulo \(4\) is \(0\) or \(1\). The right side satisfies \(4xy+3\equiv 3\pmod 4\). The left side can be \(0,1,\) or \(2\) modulo \(4\), but never \(3\). Contradiction. Therefore there are no integer solutions.
This is not pure factorisation, but it is an important insert: before searching, look for a quick modular obstruction.