Problem
NT-B1-M03-P022 The Equation \(x^2-5y^2=2\)
#22
★★★★☆ Level 4 of 5
Prove that \(x^2-5y^2=2\) has no integer solutions.
Consider the equation modulo \(5\).
Modulo \(5\), we get \(x^2\equiv2\pmod5\). But squares modulo \(5\) are only \(0,1,4\). Residue \(2\) is impossible. Therefore there are no solutions.
It looks like a Diophantine equation, but one right modulus solves it.