The factorEQ function returns the integer factorization of m in the Euclidean ring .

Given integers  and  of , with , there is an integer  such that ,  is true in . In these circumstances we say that there is a Euclidean algorithm in  and that the ring is Euclidean.

Euclidean quadratic number fields have been completely determined. They are  where d = -1, -2, -3, -7, -11, 2, 3, 5, 6, 7, 11, 13, 17, 19, 21, 29, 33, 37, 41, 57, and 73.

When ,, all integers of  have the form , where  and  are rational integers. When , all integers of  are of the form  where  and  are rational integers and of the same parity.

The answer is in the form:  such that  where  are distinct prime factors of m,  are non-negative integer numbers,  is a unit in . For real Euclidean quadratic rings, i.e.  d > 0,  is represented under the form  or  or  or  where  is the fundamental unit, and  is a positive integer.

The expand function may be applied to cause the factors to be multiplied together again.