Problem
NT-B1-M05-P005 Impossible Difference of Squares
#5
★☆☆☆☆ Level 1 of 5
Prove that the equation \(x^2-y^2=2\) has no integer solutions.
Look at squares modulo \(4\).
A square modulo \(4\) is \(0\) or \(1\). Therefore a difference of two squares modulo \(4\) can have residue \(0,1\), or \(3\), but not \(2\). The left-hand side cannot equal \(2\). No solutions exist.
This is the shortest entrance to modular obstructions in Diophantine problems.