Editing Pythagorean triple (section)

==Spinors and the modular group==
Pythagorean triples can likewise be encoded into a [[square matrix]] of the form
:<math>X = \begin{bmatrix}
c+b & a\\
a & c-b
\end{bmatrix}.
</math>
A matrix of this form is [[symmetric matrix|symmetric]]. Furthermore, the [[determinant]] of {{math|''X''}} is
:<math>\det X = c^2 - a^2 - b^2\,</math>
which is zero precisely when {{math|(''a'',''b'',''c'')}} is a Pythagorean triple. If {{math|''X''}} corresponds to a Pythagorean triple, then as a matrix it must have [[rank of a matrix|rank]] 1.

Since {{math|''X''}} is symmetric, it follows from a result in [[linear algebra]] that there is a [[column vector]] {{math|''ξ'' {{=}} [''m'' ''n'']<sup>T</sup>}} such that the [[outer product]]

{{NumBlk2|:|<math>X = 2\begin{bmatrix}m\\n\end{bmatrix}[m\ n] = 2\xi\xi^T\,</math>|1}}

holds, where the {{math|''T''}} denotes the [[matrix transpose]]. Since ξ and -ξ produce the same Pythagorean triple, the vector ξ can be considered 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 ({{EquationNote|1}}).

The [[modular group]] Γ is the set of 2×2 matrices with integer entries

:<math>A = \begin{bmatrix}\alpha&\beta\\ \gamma&\delta\end{bmatrix}</math>

with determinant equal to one: {{math|''αδ'' − ''βγ'' {{=}} 1}}. This set forms a [[group (mathematics)|group]], since the inverse of a matrix in Γ is again in Γ, as is the product of two matrices in Γ. The modular group [[Group action (mathematics)|acts]] on the collection of all integer spinors. Furthermore, the group is transitive on the collection of integer spinors with relatively prime entries. For if {{math|[''m'' ''n'']<sup>T</sup>}} has relatively prime entries, then

:<math>\begin{bmatrix}m&-v\\n&u\end{bmatrix}\begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}m\\n\end{bmatrix}</math>

where {{math|''u''}} and {{math|''v''}} are selected (by the [[Euclidean algorithm]]) so that {{math|''mu'' + ''nv'' {{=}} 1}}.

By acting on the spinor ξ in ({{EquationNote|1}}), the action of Γ goes over to an action on Pythagorean triples, provided one allows for triples with possibly negative components. Thus if {{math|''A''}} is a matrix in {{math|Γ}}, then

{{NumBlk2|:|<math>2(A\xi)(A\xi)^T = A X A^T\,</math>|2}}

gives rise to an action on the matrix {{math|''X''}} in ({{EquationNote|1}}). This does not give a well-defined action on primitive triples, since it may take a primitive triple to an imprimitive one. It is convenient at this point (per {{harvnb|Trautman|1998}}) to call a triple {{math|(''a'',''b'',''c'')}} '''standard''' if {{math|''c'' > 0}} and either {{math|(''a'',''b'',''c'')}} are relatively prime or {{math|(''a''/2,''b''/2,''c''/2)}} are relatively prime with {{math|''a''/2}} odd. If the spinor {{math|1=[''m'' ''n'']<sup>T</sup>}} has relatively prime entries, then the associated triple {{math|(''a'',''b'',''c'')}} determined by ({{EquationNote|1}}) is a standard triple. It follows that the action of the modular group is transitive on the set of standard triples.

Alternatively, restrict attention to those values of {{math|''m''}} and {{math|''n''}} for which {{math|''m''}} is odd and {{math|''n''}} is even. Let the [[subgroup]] Γ(2) of Γ be the [[Kernel (algebra)|kernel]] of the [[group homomorphism]]

:<math>\Gamma=\mathrm{SL}(2,\mathbf{Z})\to \mathrm{SL}(2,\mathbf{Z}_2)</math>

where {{math|SL(2,'''Z'''<sub>2</sub>)}} is the [[special linear group]] over the [[finite field]] {{math|'''Z'''<sub>2</sub>}} of [[modular arithmetic|integers modulo 2]]. Then Γ(2) is the group of unimodular transformations which preserve the parity of each entry. Thus if the first entry of ξ is odd and the second entry is even, then the same is true of {{math|''A''ξ}} for all {{math|''A'' ∈ Γ(2)}}. In fact, under the action ({{EquationNote|2}}), the group Γ(2) acts transitively on the collection of primitive Pythagorean triples {{harv|Alperin|2005}}.

The group Γ(2) is the [[free group]] whose generators are the matrices
:<math>U=\begin{bmatrix}1&2\\0&1\end{bmatrix},\qquad L=\begin{bmatrix}1&0\\2&1\end{bmatrix}.</math>
Consequently, every primitive Pythagorean triple can be obtained in a unique way as a product of copies of the matrices {{math|''U''}} and&nbsp;{{math|''L''}}.