Brahmagupta–Fibonacci identity

The identity [a²+b²](a²+b²)(c²+d²)=(ac−bd)²+(ad+bc)², expressing a product of two sums of two squares as another sum of two squares.

Complex multiplication: Multiplication of complex numbers, combining their real and imaginary parts by distributive algebra. The identity is the modulus-squared rule for multiplying complex numbers.
Brahmagupta: An Indian mathematician and astronomer who lived in the seventh century. His work in the Brahmasphutasiddhanta contains an early form of the identity.
Pythagorean triple: Three positive integers satisfying the equation x²+y²=z². Combining two square representations can produce new integer square representations and triples.
Euler four-square identity: An identity expressing the product of two sums of four squares as another sum of four squares. It preserves a similar product rule but uses four terms rather than two.
Complex conjugate: The complex number obtained by reversing the sign of a number’s imaginary part. Multiplying a complex number by its conjugate produces the sum of its squared coordinates.
Fibonacci: An Italian mathematician whose Liber Abaci introduced Hindu-Arabic numerals widely to medieval Europe. His work gave the identity a European transmission and its familiar alternate name.
Fermat's theorem on sums of two squares: A theorem stating that an odd prime is a sum of two squares exactly when it is congruent to 1 modulo 4. The identity helps combine prime representations into representations of composite numbers.
Difference of two squares: The factorization a²−b²=(a−b)(a+b). It factors a difference, whereas this identity composes products of sums.
Norm (algebra): A function assigning algebraic objects a scalar that is compatible with multiplication. The sum of two squares is a norm whose multiplicativity gives the identity.
Brahmasphutasiddhanta: A Sanskrit astronomical and mathematical treatise completed by Brahmagupta in 628. It presents the composition rule for sums of squares in an early documented form.

Linked from 1 page