Question

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

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

2021-06-05
$$66=2\times 3\times 3\times 37$$
$$=(1+i)(1-i)[(1+\sqrt{2i})(1-\sqrt{2i})]^{2}(1+6i)(1-6i)$$
We know $$N(a+bi)=a^{2}+b^{2}=666$$.
This looks like a circle equation where the radius is $$\sqrt{666}\approx25.81$$.
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.