TOPICS
Search

Eisenstein Prime


EisensteinPrimes

Let omega be the cube root of unity (-1+isqrt(3))/2. Then the Eisenstein primes are Eisenstein integers, i.e., numbers of the form a+bomega for a and b integers, such that a+bomega cannot be written as a product of other Eisenstein integers.

The Eisenstein primes z with complex modulus |z|<=4 are given by 1+2omega, 1-omega, -2-omega, 2omega, -2-2omega, 2, 2+3omega, 3+2omega, -2+omega, -3-omega, 1-2omega, -1-3omega, 3+4omega, 4+3omega, -3+omega, -4-omega, 1-3omega, and -1-4omega. 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. 1-omega.

2. Numbers of the form a+bomega for b=0, and a a prime congruent to 2 (mod 3).

3. Numbers of the form a+bomega or a+bomega^2 where (a+bomega)(a+bomega^2)=a^2-ab+b^2 is a prime p congruent to 1 (mod 3). Since primes of this form always have the form p=u^2+3v^2, finding the corresponding u and v gives a and b via a=u+v and b=2v.


See also

Eisenstein Integer, Eisenstein Unit, Gaussian Prime, Prime Number

Explore with Wolfram|Alpha

References

Cox, D. A. §4A in Primes of the Form x2+ny2: 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.

Referenced on Wolfram|Alpha

Eisenstein Prime

Cite this as:

Weisstein, Eric W. "Eisenstein Prime." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EisensteinPrime.html

Subject classifications