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Name: existing
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Date: Around 1995

Are there complex prime numbers? I once saw a book on the subject and I wanted to know what makes a complex number prime?

You come up with the most interesting questions. I have never heard of complex primes, but try this out:

call a + bi a complex integer if both a and b are integers.

Then a + bi would be a complex prime if there were no other complex integers that evenly divided it. Try looking in the classic work by Hardy and Wright, I think it is called The Theory Of Numbers.


The numbers Dr. Hawley alluded to, complex numbers a + bi where a and b are integers, are usually called Gaussian integers. Gaussian Integer's behave in many respects like the "rational" integers (the term often used to distinguish the "normal" integers from the Gaussian Integer's); indeed, the set of Rational Integers is a subset of the set of Gaussian Integer's. Sometimes establishing this similarity requires an extension of some concepts. For example, there are 4 Gaussian Integer counterparts to the Rational Integers 1: 1, -1, i, and -i. These are called units. If Z is a Gaussian Integer and u is a unit, then the Gaussian Integer u*Z is called an associate of Z. A Gaussian Integer is said to be prime if it is not 0 nor a unit, and if its only divisors over the Gaussian Integer's are the units and its associates. Clearly, if a Gaussian Integer is prime, so are all its associates. It can be shown that the Rational Integers primes may be used to generate all the Gaussian Integer primes: p = 2 corresponds to the Gaussian Integer prime 1+i and its associates; each odd prime p which satisfies p = 4*n + 3 for some Rational Integers n (e.g. p = 3) corresponds to the Gaussian Integer prime p + 0i (usually just written as p) and its associates; and each odd prime p which satisfies p = 4*n + 1 for some Rational Integers n (e.g. p = 5) corresponds to a Gaussian Integer prime a + bi such that a^2 + b^2 = p, and to its complex conjugate a - bi (which clearly is also prime) and their associates (e.g., p = 5 corresponds to eight Gaussian Integer primes: 2+i, its Complex Conjugate 2-i, and their associates). A theorem of Fermat says that each such p corresponds to exactly one such family of 8 Gaussian Integer primes.

Out of a dozen books on number theory, I found only one that did not refer to these numbers as Gaussian Integers (the exception, a book by the French mathematician Borel, calls them "entiers imaginary's", or imaginary inte- gers, and refers to Gaussian primes as "nombres premiers imaginaires", or imaginary prime numbers).

I did find one source that specifically uses the term "complex prime number": the "Handbook of First Complex Prime Numbers" by Kogbetliantz and Krikorian. They make no reference to Gaussian integers or primes, but their definition for a complex prime number coincides with the definition of a Gaussian prime, except they exclude the Gaussian primes corresponding to the "p = 4*n + 3" primes. I have no idea what motivated this definition, and as far as I can tell, it is not used by anyone else. Any number theorists out there?


The reason p = 4n + # p = 4n + 3 are not considered complex primes is because the they are _real_, as rcwinther pointed out. (Of course, real numbers are complex, but the a + bi primes are more interesting anyway.)

Investigating the properties of Gaussian primes is not too difficult; many texts discuss the subject (although of course a background in number theory - at least divisibility, diophantine equations, etc. is very helpful). The key is to consider the properties of the norm function N(a+bi)=a^2+b^2. Since N(y)N(z) = N(yz), thinking about the divisibility of the norm function is quite enlightening. (In particular, it provides a proof of the fact that primes of the form 4n + 3 in the regular integers cannot be factored in the complex integers either.)


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