Eisenstein prime

Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form 3n − 1. All others have an absolute value squared equal to a natural prime.

Eisenstein primes in a larger range

In mathematics, an Eisenstein prime is an Eisenstein integer

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

that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units ${\pm1, \pm ω, \pm ω 2}$ , $a + bω$ 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 $3 n - 1$ (necessarily congruent to 2 mod 3),
$| z | 2 = a 2 - ab + b 2$ 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 primes that are prime in the Gaussian integers 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 $3 n - 1$ 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 + 2 ω) 2

7 = (3 + ω)(2 - ω)

.

In general, if a natural prime p is 1 modulo 3 and can therefore be written as $p = a 2 - ab + b 2$ , 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 + 2 ω

,

6 + ω

,

7 + ω

,

7 + 3 ω

.

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 × 2^31172165 + 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.

References[]

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

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

[1]

hide v t Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
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 $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) 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 ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	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
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers

Eisenstein prime

Contents

Characterization[]

Examples[]

Large primes[]

See also[]

References[]