# Find the NORMALIZED Gaussian factorization of 666. Also, find the number of ways 666 can be written as a sum of 2 integer squares.

Find the NORMALIZED Gaussian factorization of 666. Also, find the number of ways 666 can be written as a sum of 2 integer squares.
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Benedict
$66=2×3×3×37$
$=\left(1+i\right)\left(1-i\right)\left[\left(1+\sqrt{2i}\right)\left(1-\sqrt{2i}\right){\right]}^{2}\left(1+6i\right)\left(1-6i\right)$
We know $N\left(a+bi\right)={a}^{2}+{b}^{2}=666$.
This looks like a circle equation where the radius is $\sqrt{666}\approx 25.81$.
Thus, a,b < 25.81 as a and b are to be integers.
On close observation, the values of a,b are
(22,15),(-22,15),(22,-15),(-22,-15)
(15,22),(-15,22),(15,-22),(-15,-22)
Therefore, the number of ways in which 666 can be written in sum of 2 integer squares is 8.