Brahmagupta–Fibonacci identity

In algebra, the Brahmagupta–Fibonacci identity expresses the product of two sums of two squares as a sum of two squares in two different ways. Hence the set of all sums of two squares is closed under multiplication. Specifically, the identity says $${\displaystyle (a^{2}+b^{2})(c^{2}+d^{2})=(ac\pm bd)^{2}+(ad\mp bc)^{2}}$$ with the plus-minus sign indicating that either a plus or minus may be used, with the minus-plus sign being the reverse. The equivalence of the two forms can be seen in the following example:

$${\displaystyle {\begin{aligned}(1^{2}+4^{2})(2^{2}+7^{2})&=(1\cdot 2+4\cdot 7)^{2}+(1\cdot 7-4\cdot 2)^{2},\\&\qquad =30^{2}+(-1)^{2}=\mathbf {901} ,\\&=(1\cdot 2-4\cdot 7)^{2}+(1\cdot 7+4\cdot 2)^{2},\\&\qquad =(-26)^{2}+15^{2}=\mathbf {901} .\end{aligned}}}$$

The identity is also known as the Diophantus identity, as it was first proved by Diophantus of Alexandria. It is a special case of Euler's four-square identity, and also of Lagrange's identity.

Brahmagupta proved and used a more general Brahmagupta identity, stating $${\displaystyle (a^{2}+nb^{2})(c^{2}+nd^{2})=(ac\pm nbd)^{2}+n(ad\mp bc)^{2}}$$ This shows that, for any fixed A, the set of all numbers of the form x² + Ay² is closed under multiplication.

These identities hold for all integers, as well as all rational numbers; more generally, they are true in any commutative ring. All four forms of the identity can be verified by expanding each side of the equation.