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Brahmagupta–Fibonacci identity

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In algebra, the Diophantus–Fibonacci identity, Brahmagupta-Fibonacci identity or simply Fibonacci's identity (and in fact due to Diophantus of Alexandria) 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:

Contents

( a 2 + b 2 ) ( c 2 + d 2 ) = ( a c − b d ) 2 + ( a d + b c ) 2 ( 1 ) = ( a c + b d ) 2 + ( a d − b c ) 2 . ( 2 )

Both (1) and (2) 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.

For example,

( 1 2 + 4 2 ) ( 2 2 + 7 2 ) = 26 2 + 15 2 = 30 2 + 1 2 .

The identity is a special case of Lagrange's identity. When used in conjunction with one of Fermat's theorems this proves that the product of a square and any number of primes of the form 4n + 1 is a sum of two squares.

Brahmagupta proved and used a more general identity (the Brahmagupta identity), equivalent to

( a 2 + n b 2 ) ( c 2 + n d 2 ) = ( a c − n b d ) 2 + n ( a d + b c ) 2 ( 3 ) = ( a c + n b d ) 2 + n ( a d − b c ) 2 , ( 4 )

This shows that, for any fixed n, the set of all numbers of the form x2 + n y2 is closed under multiplication.

The identity holds in the ring of integers, the ring of rational numbers and, more generally, any commutative ring (note that n could then be either an element of the ring or an ordinary integer, if multiplication by an integer is defined by repeatedly adding a ring element or its opposite).

History

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.

Analogous identities are Euler's four-square related to quaternions, and Degen's eight-square derived from the octonions which has connections to Bott periodicity. There is also Pfister's sixteen-square identity, though it is no longer bilinear.

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 + b i | | c + d i | = | ( a + b i ) ( c + d i ) |

since

| a + b i | | c + d i | = | ( a c − b d ) + i ( a d + b c ) | ,

by squaring both sides

| a + b i | 2 | c + d i | 2 = | ( a c − b d ) + i ( a d + b c ) | 2 ,

and by the definition of absolute value,

( a 2 + b 2 ) ( c 2 + d 2 ) = ( a c − b d ) 2 + ( a d + b c ) 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 + b i ) = a 2 + b 2  and  N ( c + d i ) = c 2 + d 2 ,

and also

N ( ( a + b i ) ( c + d i ) ) = N ( ( a c − b d ) + i ( a d + b c ) ) = ( a c − b d ) 2 + ( a d + b c ) 2 .

Therefore, the identity is saying that

N ( ( a + b i ) ( c + d i ) ) = N ( a + b i ) ⋅ N ( c + d i ) .

Application to Pell's equation

In its original context, Brahmagupta applied his discovery (the Brahmagupta identity) to the solution of Pell's equation, namely x2 − Ny2 = 1. Using the identity in the more general form

( x 1 2 − N y 1 2 ) ( x 2 2 − N y 2 2 ) = ( x 1 x 2 + N y 1 y 2 ) 2 − N ( x 1 y 2 + x 2 y 1 ) 2 ,

he was able to "compose" triples (x1, y1, k1) and (x2, y2, k2) that were solutions of x2 − Ny2 = k, to generate the new triple

( x 1 x 2 + N y 1 y 2 , x 1 y 2 + x 2 y 1 , k 1 k 2 ) .

Not only did this give a way to generate infinitely many solutions to x2 − Ny2 = 1 starting with one solution, but also, by dividing such a composition by k1k2, integer or "nearly integer" solutions could often be obtained. The general method for solving the Pell equation given by Bhaskara II in 1150, namely the chakravala (cyclic) method, was also based on this identity.

References

Brahmagupta–Fibonacci identity Wikipedia
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