How to Factor Gaussian Integers into Gaussian Primes

Gaussian Integer Factorization Calculator

Please enter a Gaussian integer with a modulus less than 30,000,000.
+ i

Gaussian integers are complex numbers a + bi where a and b are integers. Gaussian primes are Gaussian integers that cannot be factored into smaller Gaussian integers. For instance, 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 a 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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