A Gaussian integer is a complex number where and are integers. The Gaussian integers
are members of the imaginary quadratic field and form a ring
often denoted ,
or sometimes
(Hardy and Wright 1979, p. 179). The sum, difference, and product of two Gaussian
integers are Gaussian integers, but only if there is an such that
(1)
(Shanks 1993).
Gaussian integers can be uniquely factored in terms of other Gaussian integers (known as Gaussian primes) up to powers
of
and rearrangements.
The units of
are
and .
One definition of the norm of a Gaussian integer is its complex
modulus
(2)
Another common definition (e.g., Herstein 1975; Hardy and Wright 1979, p. 182; Artin 1991; Dummit and Foote 2004) defines the norm of a Gaussian integer to be
(3)
the square of the above quantity. (Note that the Gaussian integers form a Euclidean ring, which is what makes them particularly of interest, only under the latter
definition.) Because of the two possible definitions, caution is needed when consulting
the literature.
The probability that two Gaussian integers and are relatively prime is
Artin, M. Algebra. Englewood Cliffs, NJ: Prentice-Hall, 1991.Collins, G. E. and Johnson,
J. R. "The Probability of Relative Primality of Gaussian Integers."
Proc. 1988 Internat. Sympos. Symbolic and Algebraic Computation (ISAAC), Rome
(Ed. P. Gianni). New York: Springer-Verlag, pp. 252-258, 1989.Conway,
J. H. and Guy, R. K. "Gauss's Whole Numbers." In The
Book of Numbers. New York: Springer-Verlag, pp. 217-223, 1996.Dummit,
D. S. and Foote, R. M. Abstract
Algebra, 3rd ed. Englewood Cliffs, NJ: Prentice-Hall, 2004.Finch,
S. R. Mathematical
Constants. Cambridge, England: Cambridge University Press, 2003.Hardy,
G. H. and Wright, E. M. "The Rational Integers, the Gaussian Integers,
and the Integers of "
and "Properties of the Gaussian Integers." §12.2 and 12.6 in An
Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon
Press, pp. 178-180 and 182-183, 1979.Herstein, I. N. Topics
in Algebra, 2nd ed. New York: Springer-Verlag, 1975.Pegg, E. Jr.
"The Neglected Gaussian Integers." http://www.mathpuzzle.com/Gaussians.html.Séroul,
R. "The Gaussian Integers." §9.1 in Programming
for Mathematicians. Berlin: Springer-Verlag, pp. 225-234, 2000.Shanks,
D. "Gaussian Integers and Two Applications." §50 in Solved
and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 149-151,
1993.Sloane, N. J. A. Sequence A088454
in "The On-Line Encyclopedia of Integer Sequences."Trott,
M. The
Mathematica GuideBook for Graphics. New York: Springer-Verlag, 2004. http://www.mathematicaguidebooks.org/.
where 
and 
are integers. The Gaussian integers
are members of the imaginary quadratic field 
and form a ring
often denoted ![Z[i]](/images/equations/GaussianInteger/Inline5.svg)
,
or sometimes 
(Hardy and Wright 1979, p. 179). The sum, difference, and product of two Gaussian
integers are Gaussian integers, but 
only if there is an 
such that
and rearrangements.
![Z[i]](/images/equations/GaussianInteger/Inline10.svg)
are 
and 
.
and 
are relatively prime is
is Catalan's constant
(Pegg; Collins and Johnson 1989; Finch 2003, p. 601).
of a multiple of a Gaussian integer 
.
![RadicalBox[g, r]](/images/equations/GaussianInteger/Inline18.svg)
of the Gaussian integers for various rational
values of 
(Trott 2004, p. 24).
"
and "Properties of the Gaussian Integers." §12.2 and 12.6 in