Problem
NT-B2-M07-P008 Linear Terms in a Difference of Squares
#8
★★★☆☆ Level 3 of 5
Find all positive integers \(x>y\) satisfying \(x^2-y^2=2x+2y+21\).
Factor out \(x+y\): the left side is \((x-y)(x+y)\), and the right side contains \(2(x+y)\).
Rewrite the equation as \((x+y)(x-y)-2(x+y)=21\), hence \((x+y)(x-y-2)=21\). Let \(A=x+y\), \(B=x-y-2\). The positive factor pairs \((A,B)\) are \((1,21),(3,7),(7,3),(21,1)\). The first two give nonpositive \(y\). From \((7,3)\) we get \(x=6,y=1\), and from \((21,1)\) we get \(x=12,y=9\). The answer is \((6,1),(12,9)\).
Emphasise checking positivity after changing variables back.