Problem
ALG-B1-M09-P009 Modulo 3 obstruction
#9
★★☆☆☆ Level 2 of 5
Prove that \(x^2+1=3y\) has no integer solutions.
Which residues can a square have modulo \(3\)?
A square modulo \(3\) is \(0\) or \(1\). Then \(x^2+1\) has residue \(1\) or \(2\), while \(3y\equiv0\pmod3\). Contradiction.
A short example: a modulus can completely rule out an equation.