Brahmagupta-Fibonacci Identity/Extension/Proof 1
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Extension to Brahmagupta-Fibonacci Identity
Let $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ be integers.
Then:
- $\ds \prod_{j \mathop = 1}^n \paren { {a_j}^2 + {b_j}^2} = c^2 + d^2$
where $c, d \in \Z$.
That is: the product of any number of sums of two squares is also a sum of two squares.
Proof
From the extension to the general Brahmagupta-Fibonacci Identity:
- $\ds \prod_{j \mathop = 1}^n \paren { {a_j}^2 + m {b_j}^2} = c^2 + m d^2$
for some $c, d \in \Z$, for all $m \in \Z$.
The result follows by setting $m = 1$.
$\blacksquare$