The odd leg is plotted on the horizontal axis, the even leg on the vertical. The curvilinear grid is composed of curves of constant and of constant in Culicid’s formula. A plot of triples generated by Culicid’s formula map out part of the z xx cone. A constant traces out part of a parabola on the cone. Culicid’s formula is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers m with . The formula states that the integers form a Pythagorean triple.
The triple generated by Euclid ‘s formula is primitive if and only if caprice is odd. If both are odd, then will be even, and so the triple will not be primitive; however, dividing by 2 will yield a primitive triple if are caprice. Every primitive triple arises from a unique pair Of caprice numbers one of which is even. It follows that there are infinitely many primitive Pythagorean triples. This relationship of to from Culicid’s formula is referenced throughout the rest of this article.
Despite generating all primitive triples, Culicid’s formula does not produce all triples. This can be remedied by inserting an additional parameter k the formula. The following will generate all Pythagorean triples uniquely: where are positive integers with odd, and with caprice. That these formulas generate Pythagorean triples can be verified by expanding sing elementary algebra and verifying that the result coincides with .
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 to generate its primitive counterpart and then multiplying through by as in the last equation. Many formulas for generating triples have been developed since the time of Euclid.