# How to Factor Gaussian Integers into Gaussian Primes

*Gaussian integers* are complex numbers *a + b*i where *a* and *b* are integers. *Gaussian primes* are Gaussian integers that cannot be factored into smaller Gaussian integers. For instance, the the number 23 + 41i can be factored into

(1 - i)(1 + 2i)(2 + 3i)(1 + 4i);

each of these smaller Gaussian integers is a Gaussian prime. Factorization is unique up to multiplication by the units 1, -1, i, and -i. For example, an equivalent factorization of 23 + 41i is

(1 + i)(2 - i)(3 - 2i)(4 - i).

You can use the calculator at left to factor any Gaussian integer where *a* and *b* are not equal to zero. To factor integers and pure imaginary numbers over the Gaussian primes, use the **other calculator**.

### Gaussian Prime Factorization of a Gaussian Integer

**First**, divide out the GCD of

*a*and

*b*to form a reduced Gaussian integer.

**Next**, multiply the reduced Gaussian integer by its complex conjugate to form a regular integer. For example, with 23 + 41i we compute the product

(23 + 41i)(23 - 41i) = 2210.

**Now**, follow the method of factoring integers over the Gaussian primes outlined in this article. Continuing with the example, we obtain

2210 = (1+i)(1-i)(1+2i)(1-2i)(2+3i)(2-3i)(1+4i)(1-4i)

Keep in mind that only

*one*member of each complex conjugate pair is factor of the original number 23 + 41i. For example, 1 + 4i divides evenly into 23 + 41i, but 1 - 4i does not. To determine which one is the correct factor, you must test them both. At the end of the process you will arrive at the correct factorization.

The

**last**step is to find the Gaussian prime factorization of the GCD of

*a*and

*b*, which was factored out in the first step. Putting all the pieces together will give you the Gaussian prime factorization. You can also use the Gaussian integer factorization calculator above.

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