# Proof using primes, divisibility, and sum of squares

• ccsmarty

## Homework Statement

I have to prove or disprove the following:

Part a) If p is prime and p | (a2 + b2) and p | (c2 + d2), then p | (a2 - c2)

Part b) f p is prime and p | (a2 + b2) and p | (c2 + d2), then p | (a2 + c2)

## The Attempt at a Solution

Part a)

Since p | (a2 + b2), we have that a2 + b2 = pk, for some integer k.
Since p | (c2 + d2), we have that c2 + d2 = pt, for some integer t.

Suppose p | (a2 - c2), then we have that a2 - c2 = pr, for some integer r.

By solving for a2 and c2 in the above equations, and substitution we have that
pk - b2 - (pt - d2) = pr
pk - pt - b2 + d2 = pr
pk - pt - pr = b2 - d2
p (k - t - r) = b2 - d2
So p | (b2 - d2)

I don't know where to go from here.

I figure that once I figure out how to do this part, the second part should be very similar.
Any help would be greatly appreciated. I'm going CRAZY trying to figure this out...

Did you try looking for counterexamples before going CRAZY? That's always a good idea.