Product of Sequence of Sums of Squares equals Sum of Squares

Theorem
For all $n \in \Z_{>0}$, let $p_n = {a_n}^2 + {b_n}^2$ where $a_n$ and $b_n$ are positive integers.

Let $M \in \Z_{>0}$.

Then:
 * $\displaystyle \exists A, B \in \Z_{>0}: \prod_{k \mathop = 1}^M p_k = A^2 + B^2$

Proof
Let $z_1 = a_1 + i b_1, z_2 = a_2 + i b_2, \ldots, z_n = a_n + i b_n$.

Let $A + i B = z_1 z_2 \cdots z_M$.

Then:

As $a_1, a_2, \dotsc, a_M$ and $b_1, b_2, \dotsc, b_M$ are positive integers, then so are $A$ and $B$.

Thus: