Prove that the number x+y-1 is composite

[MM]

Let [tex]x>1[/tex] and [tex]y>1[/tex] be natural numbers such that [tex]x+y-1|x^{2}+y^{2}-1[/tex]. Prove that the number x+y-1 is composite.

[broniran]

We assume that [tex]x+y-1[/tex] is prime.Since [tex]x^2+y^2-1= (x+y-1)(x+y+1)-2xy[/tex] , we have [tex]x+y-1|2xy[/tex]. Because [tex]x+y-1[/tex] is prime (by assumption) we have [tex]x+y-1|x[/tex] or [tex]x+y-1|y[/tex] or [tex]x+y-1|2[/tex]. The first 2 cases are impossible since [tex]x+y-1>x[/tex] and [tex]x+y-1>y[/tex]. So we have [tex]x+y-1|2[/tex]-impossible