Eisenstein Prime (original) (raw)
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Let 
be the cube root of unity 
. Then the Eisenstein primes are Eisenstein integers, i.e., numbers of the form 
for 
and 
integers, such that 
cannot be written as a product of other Eisenstein integers.
The Eisenstein primes 
with complex modulus 
are given by 
, 
, 
, 
, 
, 2, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, 
, and 
. The positive Eisenstein primes with zero imaginary part are precisely the ordinary primes that are congruent to 2 (mod 3), i.e., 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, ... (OEIS A003627).
In particular, there are three classes of Eisenstein primes (Cox 1989; Wagon 1991, p. 320):
1. 
.
2. Numbers of the form 
for 
, and 
a prime congruent to 2 (mod 3).
3. Numbers of the form 
or 
where 
is a prime 
congruent to 1 (mod 3). Since primes of this form always have the form 
, finding the corresponding 
and 
gives 
and 
via 
and 
.
See also
Eisenstein Integer, Eisenstein Unit, Gaussian Prime, Prime Number
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References
Cox, D. A. §4A in Primes of the Form _x_2+n _y_2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, 1989.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.Sloane, N. J. A. Sequence A003627/M1388 in "The On-Line Encyclopedia of Integer Sequences."Wagon, S. "Eisenstein Primes." §9.8 in Mathematica in Action. New York: W. H. Freeman, pp. 319-323, 1991.
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Cite this as:
Weisstein, Eric W. "Eisenstein Prime." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EisensteinPrime.html