Problem
NT-B2-M08-P007 Another Impossible Equation
#7
★★★★☆ Level 4 of 5
Prove that \(x^2+y^2+2=3xy\) has no solutions in positive integers.
Check \(x=1,2\), then use Vieta jumping.
Let \(x\le y\) and take a minimal solution. For \(x=1\), \(y^2-3y+3=0\) has no integer roots; for \(x=2\), \(y^2-6y+6=0\) also has no integer roots. For \(x\ge 3\), the second root is \(y'=3x-y=\frac{x^2+2}{y}\), positive and integral. Since \(y\ge x\), \(y'\le x+\frac{2}{x}
Teaches checking small values before the general bound.