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Eisenstein Integer

The numbers $a+b\omega$, where

\omega\equiv {\textstyle{1\over 2}}(-1+i\sqrt{3}\,)

is one of the Roots of $z^3=1$, the others being 1 and

\omega^2={\textstyle{1\over 2}}(-1-i\sqrt{3}\,).

Eisenstein integers are members of the Quadratic Field $\Bbb{Q}(\sqrt{-3}\,)$, and the Complex Numbers ${\Bbb{Z}}[\omega]$. Every Eisenstein integer has a unique factorization. Specifically, any Nonzero Eisenstein integer is uniquely the product of Powers of $-1$, $\omega$, and the ``positive'' Eisenstein Primes (Conway and Guy 1996). Every Eisenstein integer is within a distance $\vert n\vert/\sqrt{3}$ of some multiple of a given Eisenstein integer $n$.

Dörrie (1965) uses the alternative notation

$\displaystyle J$ $\textstyle \equiv$ $\displaystyle {\textstyle{1\over 2}}(1+i\sqrt{3}\,)$ (1)
$\displaystyle O$ $\textstyle \equiv$ $\displaystyle {\textstyle{1\over 2}}(1-i\sqrt{3}\,).$ (2)

for $-\omega^2$ and $-\omega$, and calls numbers of the form $aJ+bO$ G-Number. $O$ and $J$ satisfy
$\displaystyle J+O$ $\textstyle =$ $\displaystyle 1$ (3)
$\displaystyle JO$ $\textstyle =$ $\displaystyle 1$ (4)
$\displaystyle J^2+O$ $\textstyle =$ $\displaystyle 0$ (5)
$\displaystyle O^2+J$ $\textstyle =$ $\displaystyle 0$ (6)
$\displaystyle J^3$ $\textstyle =$ $\displaystyle -1$ (7)
$\displaystyle O^3$ $\textstyle =$ $\displaystyle -1.$ (8)

The sum, difference, and products of $G$ numbers are also $G$ numbers. The norm of a $G$ number is
\end{displaymath} (9)

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


Iff $3\notdiv p+1$. These are precisely the Primes of the form $3m^2+n^2$ (Conway and Guy 1996).

See also Eisenstein Prime, Eisenstein Unit, Gaussian Integer, Integer


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 $x^2+ny^2$: 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.

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© 1996-9 Eric W. Weisstein