[Algebra] Brahmagupta–Fibonacci Identity

2021-07-29 08:30 作者:AoiSTZ23 0人读过 | 我要投稿

By: Tao Steven Zheng (郑涛)

【Problem】

The Brahmagupta–Fibonacci identity states that if two positive integers are each the sum of two Squares, then their product is the sum of two Squares. More specifically, consider four positive integers[1] $a%2C%20b%2C%20c%2C%20d$ :

$(a%5E2%2Bb%5E2%20)(c%5E2%2Bd%5E2%20)%3D(ac%E2%88%93bd)%5E2%2B(ad%C2%B1bc)%5E2$

The identity first appeared in Diophantus’ "Arithmetica" in the 3rd century AD and later reappeared in Fibonacci’s "Liber Quadratorum" (Book of Squares) in 1225 AD. In the "Brahma-sphuta-siddhanta" (Correctly Established Doctrine of Brahma), Brahmagupta generalized it to

$(a%5E2%2Bnb%5E2%20)(c%5E2%2Bnd%5E2%20)%3D(ac%E2%88%93nbd)%5E2%2Bn(ad%C2%B1bc)%5E2$

where $n$ is also a positive integer. Prove the general formula of Brahmagupta.

[1] The identity is true for real numbers a, b, c, d.

【Solution】

$(a%5E2%2Bnb%5E2%20)(c%5E2%2Bnd%5E2%20)%3Da%5E2%20c%5E2%2Bna%5E2%20d%5E2%2Bnb%5E2%20c%5E2%2Bn%5E2%20b%5E2%20d%5E2%20$

$(a%5E2%2Bnb%5E2%20)(c%5E2%2Bnd%5E2%20)%3Da%5E2%20c%5E2%2Bna%5E2%20d%5E2%2Bnb%5E2%20c%5E2%2Bn%5E2%20b%5E2%20d%5E2-2nabcd%2B2nabcd$

$(a%5E2%2Bnb%5E2%20)(c%5E2%2Bnd%5E2%20)%3D(a%5E2%20c%5E2%E2%88%932nabcd%2Bn%5E2%20b%5E2%20d%5E2%20)%2B(na%5E2%20d%5E2%C2%B12abcd%2Bnb%5E2%20c%5E2%20)$