# system of binomial factors

A system of any number of binomial factors each the sum of two squares being supposed multiplied together , show , by any method, that their product is the sum of two squares.

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Please share your thoughts so far :) –  Alice Jul 6 '14 at 16:24

## 1 Answer

Hint: $(a^2+b^2)(p^2+q^2)=(ap-bq)^2+(aq+bp)^2$. Using this identity repeatedly will do the job.

The identity can be verified most simply by expanding each side. It is sometimes called the Brahmagupta-Fibonacci Identity.

Remark: Brahmagupta actually used a more general identity, in a study of the solutions of the so-called Pell equation. He did not call it the Pell equation, since his work was done about $1000$ years before Pell lived.

If you are familiar with complex numbers, you can note that $(a+bi)(p+qi)=(ap-bq) +i(aq+bp)$. Thus the identity can be viewed as saying that the norm of a product is the product of the norms.

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