− (2×4) + (2×5), (2×3) − 4 + (2×5), (2×3) − (2×4) + (3×5)) = (5, 12, 13), and similarly and produce the triples (21, 20, 29) and (15, 8, 17). The linear transformations T1, T2, and T3 have a geometric interpretation in the language of quadratic forms. They are closely related to (but are not equal to) reflections generating the orthogonal group of over the integers. Relation to Gaussian integers Alternatively, Euclid's formulae can be analyzed and proved using the Gaussian integers. Gaussian integers are complex numbers of the form , where and are ordinary integers and is the square root of negative one. The units of Gaussian integers are ±1 and ±i. The ordinary integers are called the rational integers and denoted as ''. The Gaussian integers are denoted as . The right-hand side of the Pythagorean theorem may be factored in Gaussian integers: A primitive Pythagorean triple is one in which and are coprime, i.e., they share no prime factors in the integers. For such a triple, either or is even, and the other is odd; from this, it follows that is also odd. The two factors and of a primitive Pythagorean triple each equal the square of a Gaussian integer. This can be proved using the property that every Gaussian integer can be factored uniquely into Gaussian primes up to units. (This unique factorization follows from the fact that, roughly speaking, a version of the Euclidean algorithm can be defined on them.) The proof has three steps. First, if and share no prime factors in the integers, then they also share no prime factors in the Gaussian integers. (Assume and with Gaussian integers , and and not a unit. Then and lie on the same line through the origin. All Gaussian integers on such a line are integer multiples of some Gaussian integer . But then the integer gh ≠ ±1 divides both and .) Second, it follows that and likewise share no prime factors in the Gaussian integers. For if they did, then their common divisor would also divide and . Since and are coprime, that implies that divides . From the formula , that in turn would imply that is even, contrary to the hypothesis of a primitive Pythagorean triple. Third, since is a square, every Gaussian prime in its factorization is doubled, i.e., appears an even number of times. Since and share no