For example:
a b c 3 4 5 5 12 13 6 8 10 7 24 25 8 15 17 9 12 15
If (a,b,c) is a Pythagorean triple so is (da,db,dc) for any positive integer d. A Pythagorean triple is said to be primitive if a, b and c have no common divisor.
If m > n are positive integers, then
is a Pythagorean triple. It is primitive if and only if m and n are coprime and one of them is even (if both n and m are odd, then a, b, and c will be even, and so the Pythagorean triple will not be primitive). Not every Pythagorean triple can be generated in this way, but every primitive triple (possibly after exchanging a and b) arises in this fashion from a unique pair of coprime numbers m > n. This shows that there are infinitely many primitive Pythagorean triples.
Fermat's Last Theorem states that nontrivial triples analogous to Pythagorean triples but with exponents higher than 2 don't exist.
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