count is The area (by Pick's theorem equal to one less than the interior lattice count plus half the boundary lattice count) equals . The first occurrence of two primitive Pythagorean triples sharing the same area occurs with triangles with sides and common area 210 . The first occurrence of two primitive Pythagorean triples sharing the same interior lattice count occurs with and interior lattice count 2287674594 . Three primitive Pythagorean triples have been found sharing the same area: , , with area 13123110. As yet, no set of three primitive Pythagorean triples have been found sharing the same interior lattice count. Enumeration of primitive Pythagorean triples By Euclid's formula all primitive Pythagorean triples can be generated from integers and with , odd and . Hence there is a 1 to 1 mapping of rationals (in lowest terms) to primitive Pythagorean triples where is in the interval and odd. The reverse mapping from a primitive triple where to a rational is achieved by studying the two sums and . One of these sums will be a square that can be equated to and the other will be twice a square that can be equated to . It is then possible to determine the rational . In order to enumerate primitive Pythagorean triples the rational can be expressed as an ordered pair and mapped to an integer using a pairing function such as Cantor's pairing function. An example can be seen at . It begins and gives rationals these, in turn, generate primitive triples Spinors and the modular group Pythagorean triples can likewise be encoded into a square matrix of the form A matrix of this form is symmetric. Furthermore, the determinant of is which is zero precisely when is a Pythagorean triple. If corresponds to a Pythagorean triple, then as a matrix it must have rank 1. Since is symmetric, it follows from a result in linear algebra that there is a column vector such that the outer product holds, where the denotes the matrix transpose. The vector ξ is called a spinor (for the Lorentz group SO(1, 2)). In abstract terms, the Euclid formula means that each primitive Pythagorean triple can be written as the outer product with itself of a spinor with integer entries, as in (). The modular group Γ is the set of 2×2 matrices with integer entries with determinant equal to one: . This set