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.
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.