Problem
ALG-B1-M09-P018 Modulo 4 obstruction
#18
★★★★☆ Level 4 of 5
Prove that \(x^2+y^2=4z+3\) has no integer solutions.
Which residues do squares give modulo \(4\)?
A square modulo \(4\) is \(0\) or \(1\). Thus the sum of two squares modulo \(4\) can be \(0,1,2\), but the right-hand side \(4z+3\) has residue \(3\). Contradiction.
One of the most common modular obstructions.