Each of x and y is a positive integer such that: x² – 5y² = 1.

Prove that x*y is always divisible by 12.

(In reply to Possible solution by broll)

Your elementary method is really cool, but since I dont see why x is invariably odd I just went and used the recurrence.

The fundamental (x1,y1) = (9,4) works

Forgive my lack of subscripts
xk+1 = x1xk + ny1yk
yk+1 = x1yk + y1xk

so for this we have
xk+1 = 9xk + 20yk
yk+1 = 9yk + 4xk

and the product xk+1*yk+1 simplifies to
36xk^2 + 161xkyk + 180yk^2

since 36, xkyk, and 180 are all divisible by 12 so does this sum.
So we have a recursive proof.

Posted by Jer on 2010-12-03 17:52:28