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
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.)
Click here to return to the Mathematics Archives
Update: June 2012