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 2 of 3 |

If a and b are coprime then a+b is also coprime to a and b.

Then any factor of a+b is coprime to a and b.

Let f be some common factor of a+b and a^2+b^2.

Then f is also a factor of (a+b)^2.

Then f is also a factor of (a+b)^2 - a^2+b^2 = 2ab.

Then f is a common factor of a+b and 2ab.

Since any factor of a+b is coprime to a and b, then f is a factor of 2.

So f is at most 2. QED.

Posted by Brian Smith on 2021-08-05 11:42:52

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