Problem
ALG-B1-M09-P019 Form equals one
#19
★★★★☆ Level 4 of 5
Find all integer pairs \((x,y)\) such that \(x^2-xy+y^2=1\).
Multiply by \(4\): \((2x-y)^2+3y^2=4\).
We have \((2x-y)^2+3y^2=4\). Thus \(y\in\{-1,0,1\}\). If \(y=0\), then \(x=\pm1\). If \(y=1\), then \((2x-1)^2=1\), so \(x=0\) or \(1\). If \(y=-1\), then \((2x+1)^2=1\), so \(x=0\) or \(-1\). Answer: \((1,0),(-1,0),(0,1),(1,1),(0,-1),(-1,-1)\).
The quadratic form turns an infinite search into a small check.