perplexus.info :: Numbers : A coprime couple

A coprime couple (Posted on 2021-08-05)

Prove the following:
For any two positive coprime numbers a and b
no common divider greater than 2 exists
for a+b & a²+b²

No Solution Yet Submitted by Ady TZIDON

Rating: 4.0000 (1 votes)

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Solution

| Comment 1 of 3

Say a^2 + b^2 and a + b were each divisible by some prime number P. So

a^2 + b^2 = PX

a + b = PY

for some integers, X,Y

Since one of a, b is even, one odd, P is not 2.

Then 2ab = (PY^2) - PX = P(PY - X) is also divisible by P. So P must be one of the prime factors occurring either in a or b. Assume, without losing generality, that a = PZ then

b = PY - a = P(Y - Z), so that both a and b share the same common factor, namely P. A contradiction.

Posted by FrankM on 2021-08-05 11:19:23

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