Eisenstein prime

In mathematics, an Eisenstein prime is an Eisenstein integer

\( {\displaystyle z=a+b\,\omega ,\quad {\text{where}}\quad \omega =e^{\frac {2\pi i}{3}},} \)

that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units {±1, ±ω, ±ω2}, a + bω itself and its associates.

The associates (unit multiples) and the complex conjugate of any Eisenstein prime are also prime.

Characterization

An Eisenstein integer z = a + bω is an Eisenstein prime if and only if either of the following (mutually exclusive) conditions hold:

z is equal to the product of a unit and a natural prime of the form 3n − 1 (necessarily congruent to 2 mod 3), |z|² = a² − ab + b² is a natural prime (necessarily congruent to 0 or 1 mod 3).

It follows that the square of the absolute value of every Eisenstein prime is a natural prime or the square of a natural prime.

In base 12 (written with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B), the natural Eisenstein primes are exactly the natural primes ending with 5 or B (i.e. the natural primes congruent to 2 mod 3). The natural Gaussian primes are exactly the natural primes ending with 7 or B (i.e. the natural primes congruent to 3 mod 4).

Examples

The first few Eisenstein primes that equal a natural prime 3n − 1 are:

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ... (sequence A003627 in the OEIS).

Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes: they admit nontrivial factorizations in Z[ω]. For example:

3 = −(1 + 2ω)²

7 = (3 + ω)(2 − ω).

In general, if a natural prime p is 1 modulo 3 and can therefore be written as p = a² − ab + b², then it factorizes over Z[ω] as

p = (a + bω)((a − b) − bω).

Some non-real Eisenstein primes are

2 + ω, 3 + ω, 4 + ω, 5 + 2ω, 6 + ω, 7 + ω, 7 + 3ω.

Up to conjugacy and unit multiples, the primes listed above, together with 2 and 5, are all the Eisenstein primes of absolute value not exceeding 7.

Large primes

As of September 2019, the largest known (real) Eisenstein prime is the ninth largest known prime 10223 × 231172165 + 1, discovered by Péter Szabolcs and PrimeGrid.[1] All larger known primes are Mersenne primes, discovered by GIMPS. Real Eisenstein primes are congruent to 2 mod 3, and all Mersenne primes greater than 3 are congruent to 1 mod 3; thus no Mersenne prime is an Eisenstein prime.
See also

Gaussian prime

References

Chris Caldwell, "The Top Twenty: Largest Known Primes" from The Prime Pages. Retrieved 2019-09-18.

Prime number classes
By formula

Fermat (2^2ⁿ + 1) Mersenne (2^p − 1) Double Mersenne (2^{2^p−1} − 1) Wagstaff (2^p + 1)/3 Proth (k·2ⁿ + 1)

Factorial (n! ± 1) Primorial (p_n# ± 1) Euclid (p_n# + 1) Pythagorean (4n + 1) Pierpont (2^m·3ⁿ + 1)

Quartan (x⁴ + y⁴) Solinas (2^m ± 2ⁿ ± 1) Cullen (n·2ⁿ + 1) Woodall (n·2ⁿ − 1) Cuban (x³ − y³)/(x − y)

Carol (2ⁿ − 1)² − 2 Kynea (2ⁿ + 1)² − 2 Leyland (x^y + y^x) Thabit (3·2ⁿ − 1) Williams ((b−1)·bⁿ − 1)

Mills (⌊A^3ⁿ⌋)

By integer sequence

Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin

By property

Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient

Base-dependent

Palindromic Emirp Repunit (10n − 1)/9 Permutable Circular Truncatable Minimal Weakly Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic

Patterns

Twin (p, p + 2) Bi-twin chain (n − 1, n + 1, 2n − 1, 2n + 1, …) Triplet (p, p + 2 or p + 4, p + 6) Quadruplet (p, p + 2, p + 6, p + 8) k−Tuple Cousin (p, p + 4) Sexy (p, p + 6) Chen Sophie Germain/Safe (p, 2p + 1) Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...) Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) Balanced (consecutive p − n, p, p + n)

By size

Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known

Complex numbers

Eisenstein prime Gaussian prime

Composite numbers

Pseudoprime
Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious