Brahmagupta–Fibonacci identity

The Brahmagupta identity, also known as Brahmagupta - Fibonacci identity or Fibonacci identity is an identity in elementary algebra. Despite their name, their first known use is not due to Brahmagupta or Fibonacci, singles can be found in a work of Diophantus of Alexandria ( Arithmetica (III, 19) ).

Identity equation

The identity describes how to re- enable the product of two sums, consisting written as the sum of two other squares of two square numbers.

A numerical example:

As a direct consequence of the identity it follows that the set of sums of two squares is closed under multiplication.

Brahmagupta itself has proved a more general result and used the equivalent of

Is. As a consequence it follows that the amount of the number of the mold is completed with respect to the multiplication.

Historical

The latter identity goes back to the Indian mathematician and astronomer Brahmagupta ( 598-668 ) and finds himself in his work Brahmasphutasiddhanta from the year 628 This was first translated into Arabic by Muhammad al - Fazari from Sanskrit; to 1128 then was a translation into Latin from the Arabic version. Later, the (former) Diophantus identity is also described in Fibonacci's Liber Quadratorum of 1225.

Extensions

Brahmagupta - Fibonacci identity is a two-square identity, which can be attributed to four, eight, sixteen, and adding more squares:

• Brahmagupta - Fibonacci identity: (x ₁ ² x ₂ ² ) · ( y ₁ ₂ ² y ² ) = z ₁ ₂ ² z ²
• Euler's four -square identity: (x ₁ ² x ₂ ² x ₃ ² x ₄ ² ) · ( y ₁ ² y ₂ ² y ₃ ² y ₄ ² ) = z ₁ ² z ₂ ² z ₃ ² z ₄ ²
• Degens eight -square identity: (x ₁ ² x ₂ ² x ₃ ² ... x ₈ ² ) · ( y ₁ ² y ₂ ² y ₃ ² ... y ₈ ² ) = z ₁ ² z ₂ ² z ₃ ² ... z ₈ ²
• Pfister Sixteen -squares identity: (x ₁ ² x ₂ ² x ₃ ² ... x ₁ ₆ ² ) · ( y ₁ ² y ₂ ² y ₃ ² ... y ₁ ₆ ² ) = ( z ₁ ² z ₂ ² z ₃ ² ... z ₁ ₆ ²)

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

The two squares identity is associated with the complex numbers, the four squares identity to quaternions, the eight squares identity with octonions.