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
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...