Problem
NT-B2-M07-P015 Parametrisation by the Difference
#15
★★★★☆ Level 4 of 5
Find all positive integer pairs \((x,y)\) satisfying \(x^2+y^2=2xy+x+y\).
Move \(2xy\) to the left: \((x-y)^2\) will appear.
The equation is equivalent to \((x-y)^2=x+y\). Let \(d=|x-y|\). Then \(x+y=d^2\). If \(x\ge y\), then \(x-y=d\), hence \(x=\frac{d^2+d}{2}\), \(y=\frac{d^2-d}{2}\). Positivity requires \(d\ge 2\). Conversely, every \(d\ge 2\) gives positive integers satisfying the equation. Including the swapped order, the answer is \(\left(\frac{d(d+1)}{2},\frac{d(d-1)}{2}\right)\) and its swap, where \(d=2,3,4,\ldots\).
The answer is infinite; students should not attempt a finite search without a reason.