The numbers , where

is one of the Roots of , the others being 1 and

Eisenstein integers are members of the Quadratic Field , and the Complex Numbers . Every Eisenstein integer has a unique factorization. Specifically, any Nonzero Eisenstein integer is uniquely the product of Powers of , , and the ``positive'' Eisenstein Primes (Conway and Guy 1996). Every Eisenstein integer is within a distance of some multiple of a given Eisenstein integer .

Dörrie (1965) uses the alternative notation

(1) | |||

(2) |

for and , and calls numbers of the form

(3) | |||

(4) | |||

(5) | |||

(6) | |||

(7) | |||

(8) |

The sum, difference, and products of numbers are also numbers. The norm of a number is

(9) |

The analog of Fermat's Theorem for Eisenstein integers is that a Prime Number can be written in the
form

Iff . These are precisely the Primes of the form (Conway and Guy 1996).

**References**

Conway, J. H. and Guy, R. K. *The Book of Numbers.* New York: Springer-Verlag, pp. 220-223, 1996.

Cox, D. A. §4A in *Primes of the Form : Fermat, Class Field Theory and Complex Multiplication.*
New York: Wiley, 1989.

Dörrie, H. ``The Fermat-Gauss Impossibility Theorem.'' §21 in
*100 Great Problems of Elementary Mathematics: Their History and Solutions.* New York: Dover, pp. 96-104, 1965.

Guy, R. K. ``Gaussian Primes. Eisenstein-Jacobi Primes.'' §A16 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 33-36, 1994.

Riesel, H. Appendix 4 in *Prime Numbers and Computer Methods for Factorization, 2nd ed.* Boston, MA: Birkhäuser, 1994.

Wagon, S. ``Eisenstein Primes.'' *Mathematica in Action.* New York: W. H. Freeman, pp. 278-279, 1991.

© 1996-9

1999-05-25