Brahmagupta–Fibonacci identity

In algebra, Brahmagupta's identity, also sometimes called Fibonacci's identity, implies that the product of two sums 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:

${\begin{aligned}\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)&{}=\left(ac-bd\right)^{2}+\left(ad+bc\right)^{2}\ \qquad \qquad (1)\\&{}=\left(ac+bd\right)^{2}+\left(ad-bc\right)^{2}.\qquad \qquad (2)\end{aligned}}$

For example,

$(1^{2}+4^{2})(2^{2}+7^{2})=30^{2}+1^{2}=26^{2}+15^{2}.\,$

Both (1) and (2) can be verified by expanding each side of the equation. Also, (2) can be obtained from (1) by changing b to −b.

This identity holds for any real numbers, and more generally for members of any commutative ring.

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 times any number of primes of the form 4n+1 is also a sum of two squares.

History

The identity was discovered by Brahmagupta (598-668), an Indian mathematician and astronomer. It was later translated to Arabic and Persian, and then translated to Latin by Leonardo of Pisa (1170-1250) also known as Fibonacci. It appeared in Fibonacci's Book of Squares in 1225. It may also have been known to Diophantus in the 3rd century.

Related identities

Euler's four-square identity is an analogous identity involving four squares instead of two that is related to quaternions. There is a similar eight-square identity derived from the Cayley numbers which has connections to Bott periodicity.

Relation to complex numbers

If a, b, c, and d are real numbers, this identity is equivalent to the multiplication property for absolute values of complex numbers namely that:

$|a+bi||c+di|=|(a+bi)(c+di)|\,$

since

$|a+bi||c+di|=|(ac-bd)+i(ad+bc)|,\,$

by squaring both sides

$|a+bi|^{2}|c+di|^{2}=|(ac-bd)+i(ad+bc)|^{2},\,$

and by the definition of absolute value,

$(a^{2}+b^{2})(c^{2}+d^{2})=(ac-bd)^{2}+(ad+bc)^{2}.\,$

Interpretation via norms

In the case that the variables a, b, c, and d are rational numbers, the identity may be interpreted as the statement that the norm in the field Q(i) is multiplicative. That is, we have

$N(a+bi)=a^{2}+b^{2}\,$ and $N(c+di)=c^{2}+d^{2},\,$

and also

$N((a+bi)(c+di))=N((ac-bd)+i(ad+bc))=(ac-bd)^{2}+(ad+bc)^{2}.\,$

Therefore the identity is saying that

$N((a+bi)(c+di))=N(a+bi)\cdot N(c+di).\,$

See also

• Brahmagupta
• Brahmagupta matrix
• Complex numbers
• Indian mathematics
• List of Indian mathematicians

External links

• Brahmagupta Identity on MathWorld