Brahmagupta-Fibonacci identity
In algebra, the Brahmagupta–Fibonacci identity 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, it states
For example,
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 identity (the Brahmagupta identity), equivalent to
This shows that, for any fixed A, the set of all numbers of the form x2 + A y2 is closed under multiplication.
The identity holds in the ring of integers, the ring of rational numbers and, more generally, any commutative ring. All four forms of the identity 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, and likewise with (3) and (4).
The identity is actually first found in Diophantus' Arithmetica (III, 19), of the third century A.D. 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.
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