Brahmagupta identity

The Brahmagupta identity , as Brahmagupta-Fibonacci identity or Fibonacci identity known, is an identity in elementary algebra . Despite its name, its first known use does not go back to Brahmagupta or Fibonacci , but can be found in a work by Diophantos of Alexandria ( Arithmetica (III, 19)).

Identity equation

Identity describes how the product of two sums, each consisting of two square numbers, can be represented again as the sum of two other square numbers.

${\displaystyle \left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)}$ ${\displaystyle =\left(ac+bd\right)^{2}+\left(ad-bc\right)^{2}}$ ${\displaystyle =\left(ac-bd\right)^{2}+\left(ad+bc\right)^{2}}$

A numerical example:

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

As a direct consequence of the identity it follows that the set of the sums of two square numbers is closed with respect to the multiplication.

Brahmagupta himself has proven and used a more general result that is equivalent to

${\displaystyle {\begin{aligned}\left(a^{2}+nb^{2}\right)\left(c^{2}+nd^{2}\right)&{}=\left(ac-nbd\right)^{2}+n\left(ad+bc\right)^{2}\\&{}=\left(ac+nbd\right)^{2}+n\left(ad-bc\right)^{2}\end{aligned}}}$

is. It follows that the set of numbers is closed by the form with respect to multiplication. ${\displaystyle x^{2}+ny^{2}}$

Historical

The latter identity goes back to the Indian mathematician and astronomer Brahmagupta (598–668) and can be found in his work Brahmasphutasiddhanta from the year 628. This was first translated from Sanskrit into Arabic by Muhammad al-Fazari ; around 1128 a translation into Latin was made from the Arabic version. Later, the (earlier) Diophantos identity was also described in Fibonacci's Liber Quadratorum of 1225.

Extensions

Brahmagupta – Fibonacci identity is a two-square identity that can be expanded to four, eight, sixteen and more squares:

• Brahmagupta – Fibonacci identity: (x₁² + x₂²) · (y₁² + y₂²) = z₁² + z₂²
• Euler's four-square identity: (x₁² + x₂² + x₃² + x₄²) · (y₁² + y₂² + y₃² + y₄²) = z₁² + z₂² + z₃² + z₄²
• Degen's eight-square identity: (x₁² + x₂² + x₃² +… + x₈²) · (y₁² + y₂² + y₃² +… + y₈²) = z₁² + z₂² + z₃² +… + z₈²
• Pfister's sixteen-squares identity: (x₁² + x₂² + x₃² +… + x₁₆²) · (y₁² + y₂² + y₃² +… + y₁₆²) = (z₁² + z₂² + z₃² +… + z₁₆²)

Pfister proved in 1967 that, in principle, two (2ⁿ) identities can be found for all powers.

The two-squares identity is in connection with the complex numbers , the four-squares identity with the quaternions , the eight-square identity with the octonions , see the square theorem .

Individual evidence

1. ↑ George G. Joseph (2000). The Crest of the Peacock , p. 306. Princeton University Press . ISBN 0-691-00659-8 . (engl.)
2. ↑ Eric W. Weisstein : Fibonacci Identity . In: MathWorld (English).
3. ↑ Eric W. Weisstein : Euler Four-Square Identity . In: MathWorld (English).
4. ↑ ^{a b} Eric W. Weisstein : Degen's Eight-Square Identity . In: MathWorld (English).
5. ↑ Tito Piezas III: Pfister's 16-square Identity

Web links

• Eric W. Weisstein : Fibonacci Identity . In: MathWorld (English).
• Brahmagupta identity at PlanetMath (English)
• Brahmagupta-Fibonacci Identity