In algebra, the Brahmagupta–Fibonacci identity or simply Fibonacci's identity (and in fact due to Diophantus of Alexandria) says that the product of two sums each of two squares is itself a sum of two squares. In other words, the set of all sums of two squares is closed under multiplication. Specifically:
showing that the set of all numbers of the form is closed under multiplication.
Both (1) and (2) can be verified by expanding each side of the equation. Also, (2) can be obtained from (1), or (1) from (2), by changing b to −b.
In the integer case this identity finds applications in number theory for example when used in conjunction with one of Fermat's theorems it proves that the product of a square and any number of primes of the form 4n + 1 is also a sum of two squares.
The identity is actually first found in Diophantus' Arithmetica (III, 19), of the third century BC. It was rediscovered by Brahmagupta (598–668), an Indian mathematician and astronomer, who generalized it (to the Brahmagupta identity) and used it in his study of what is now called Pell's equation. His Brahmasphutasiddhanta was translated from Sanskrit into Arabic by Mohammad al-Fazari, and was subsequently translated into Latin in 1126. The identity later appeared in Fibonacci's Book of Squares in 1225.
Analogous identities are Euler's four-square related to quaternions, and Degen's eight-square derived from the Cayley numbers which has connections to Bott periodicity. There is also Pfister's sixteen-square identity, though it is no longer bilinear.
Relation to complex numbers
by squaring both sides
and by the definition of absolute value,
Interpretation via norms
Therefore the identity is saying that
Application to Pell's equation
he was able to "compose" triples (x1, y1, k1) and (x2, y2, k2) that were solutions of x2 − Ny2 = k, to generate the new triple
Not only did this give a way to generate infinitely many solutions to x2 − Ny2 = 1 starting with one solution, but also, by dividing such a composition by k1k2, integer or "nearly integer" solutions could often be obtained. The general method for solving the Pell equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was also based on this identity.