Twin Primes Problem

**BG5965** · July 10th 2009, 08:46 AM

Prime numbers $x$ and $y$ are called twin primes, if $y = x + 2$ . Prove that the numbers $x^4 + 4$ and $y^4 + 4$ are never relatively prime, if $x$ and $y$ are twin primes?

I'm not sure how to correctly approach this problem, but I've tried a few things and I think that $mod 4$ is involved somehow.

Please help, BG.

**PaulRS** · July 11th 2009, 04:45 AM

Try modulo 5.

- remember to check the cases that involve 5 (as one of the primes) separately-

**aidan** · July 11th 2009, 07:34 AM

The approach I use:
(and I'm guessing that you've done this.)

$y^4 + 4$ Eqn1

$y = x + 2$ Eqn2

$y^4 + 4$ = $(x+2)^4 + 4$ Eqn3

Expand the RHS and notice the difference between Eqn1 and Eqn3
I do not know if there is a "correct" approach.

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