. Then . Since is rational, we set it equal to in lowest terms. Thus , being the reciprocal of . Then solving for and gives As is fully reduced, and are coprime, and they cannot both be even. If they were both odd, the numerator of would be a multiple of 4 (because an odd square is congruent to 1 modulo 4), and the denominator 2mn would not be a multiple of 4. Since 4 would be the minimum possible even factor in the numerator and 2 would be the maximum possible even factor in the denominator, this would imply to be even despite defining it as odd. Thus one of and is odd and the other is even, and the numerators of the two fractions with denominator 2mn are odd. Thus these fractions are fully reduced (an odd prime dividing this denominator divides one of and but not the other; thus it does not divide ). One may thus equate numerators with numerators and denominators with denominators, giving Euclid's formula with and coprime and of opposite parities. A longer but more commonplace proof is given in Maor (2007) and Sierpiński (2003). Another proof is given in , as an instance of a general method that applies to every homogeneous Diophantine equation of degree two. Interpretation of parameters in Euclid's formula Suppose the sides of a Pythagorean triangle have lengths , , and , and suppose the angle between the leg of length and the hypotenuse of length is denoted as . Then and the full-angle trigonometric values are , , and . A variant The following variant of Euclid's formula is sometimes more convenient, as being more symmetric in and (same parity condition on and ). If and are two odd integers such that , then are three integers that form a Pythagorean triple, which is primitive if and only if and are coprime. Conversely, every primitive Pythagorean triple arises (after the exchange of and , if is even) from a unique pair of coprime odd integers. Elementary properties of primitive Pythagorean triples General properties The properties of a primitive Pythagorean triple with (without specifying which of or is even and which is odd) include: is always a perfect square. As it is only a necessary condition but not a sufficient one, it can be used in checking if a given triple of numbers is not a Pythagorean