## Eisenstein Integer

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 G-Number. and satisfy
 (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.