When Carl Friedrich Gauss first introduced the notion of complex integers "Z"["i"], he observed that the ordinary prime numbers may factor further in this new set of integers. In fact, if a prime "p" is congruent to 1 mod 4, then it factors into a product of two distinct prime gaussian integers, or "splits completely"; if "p" is congruent to 3 mod 4, then it remains prime, or is "inert"; and if "p" is 2 then it becomes a product of the square of the prime "(1+i)" and the invertible gaussian integer "-i"; we say that 2 "ramifies". For instance,