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In algebra, the 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: : 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, : 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 4''n'' + 1 is a sum of two squares. Brahmagupta proved and used a more general identity (the Brahmagupta identity), equivalent to : This shows that, for any fixed ''n'', the set of all numbers of the form ''x''2 + ''n'' ''y''2 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.〔George G. Joseph (2000). ''The Crest of the Peacock'', p. 306. Princeton University Press. ISBN 0-691-00659-8.〕 The identity later appeared in Fibonacci's ''Book of Squares'' in 1225. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Brahmagupta–Fibonacci identity」の詳細全文を読む スポンサード リンク
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