triple when and are coprime. Every primitive triple arises (after the exchange of and , if is even) from a unique pair of coprime numbers , , one of which is even. It follows that there are infinitely many primitive Pythagorean triples. This relationship of , and to and from Euclid's formula is referenced throughout the rest of this article. Despite generating all primitive triples, Euclid's formula does not produce all triples—for example, (9, 12, 15) cannot be generated using integer and . This can be remedied by inserting an additional parameter to the formula. The following will generate all Pythagorean triples uniquely: where , , and are positive integers with , and with and coprime and not both odd. That these formulas generate Pythagorean triples can be verified by expanding using elementary algebra and verifying that the result equals . Since every Pythagorean triple can be divided through by some integer to obtain a primitive triple, every triple can be generated uniquely by using the formula with and to generate its primitive counterpart and then multiplying through by as in the last equation. Choosing and from certain integer sequences gives interesting results. For example, if and are consecutive Pell numbers, and will differ by 1. Many formulas for generating triples with particular properties have been developed since the time of Euclid. Proof of Euclid's formula That satisfaction of Euclid's formula by a, b, c is sufficient for the triangle to be Pythagorean is apparent from the fact that for positive integers and , , the , , and given by the formula are all positive integers, and from the fact that A proof of the necessity that a, b, c be expressed by Euclid's formula for any primitive Pythagorean triple is as follows. All such primitive triples can be written as where and , , are coprime. Thus , , are pairwise coprime (if a prime number divided two of them, it would be forced also to divide the third one). As and are coprime, at least one of them is odd, so we may suppose that is odd, by exchanging, if needed, and . This implies that is even and is odd (if were odd, would be even, and would be a multiple of 4, while would be congruent to 2 modulo 4, as an odd square is congruent to 1 modulo 4). From we obtain and hence