Problem
NT-B2-M08-P008 All Solutions of \(x^2+y^2+1=3xy\)
#8
★★★★☆ Level 4 of 5
Describe all positive integer solutions of \(x^2+y^2+1=3xy\).
Show that every non-base solution descends by a Vieta jump.
Let \(x\le y\). For \(x=1\), \(y^2-3y+2=0\), so \(y=1\) or \(2\). For \(x>1\), the second root is \(y'=3x-y=\frac{x^2+1}{y}\). It is positive and integral. Since \(y\ge x\), \(y'\le x+\frac{1}{x}
This is the central training problem of the module.