
Twin Primes Problem
Prime numbers $\displaystyle x$ and $\displaystyle y$ are called twin primes, if $\displaystyle y = x + 2$. Prove that the numbers $\displaystyle x^4 + 4$ and $\displaystyle y^4 + 4$ are never relatively prime, if $\displaystyle x$ and $\displaystyle y$ are twin primes?
I'm not sure how to correctly approach this problem, but I've tried a few things and I think that $\displaystyle mod 4$ is involved somehow.
Please help, BG.

Try modulo 5. (Wink)  remember to check the cases that involve 5 (as one of the primes) separately

The approach I use:
(and I'm guessing that you've done this.)
$\displaystyle y^4 + 4$ Eqn1
$\displaystyle y = x + 2$ Eqn2
$\displaystyle y^4 + 4$ = $\displaystyle (x+2)^4 + 4$ Eqn3
Expand the RHS and notice the difference between Eqn1 and Eqn3
I do not know if there is a "correct" approach.