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)

I agree with Jer; your 'elementary' method is interesting and worth perfecting, but the 'oddness' of x still needs to be proven. Would this do the trick?...

Assuming x is even, write x = 2X, so that x² - 5y² = 1 becomes 4X² - 5y² = 1.

Then working modulo 4, this gives        y² = 3 (mod 4).

Now, squares can only take the values 0 or 1 (mod 4), so we have a contradiction, proving that if a solution exists for this equation, then x is odd.

Edited on December 3, 2010, 9:28 pm

Posted by Harry on 2010-12-03 21:26:03