mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an Eisenstein prime is an Eisenstein integer :

z = a + b\,\omega, \quad \text \quad \omega = e^,

that is

irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...

(or equivalently

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

) in the ring-theoretic sense: its only Eisenstein

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

s are the

unit Unit may refer to: Arts and entertainment * UNIT, a fictional military organization in the science fiction television series ''Doctor Who'' * Unit of action, a discrete piece of action (or beat) in a theatrical presentation Music * ''Unit'' (alb ...

s , itself and its associates. The associates (unit multiples) and the

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...

of any Eisenstein prime are also prime.

Characterization

An Eisenstein integer is an Eisenstein prime if and only if either of the following (mutually exclusive) conditions hold: # is equal to the product of a unit and a natural prime of the form (necessarily congruent to ), # is a natural prime (necessarily congruent to 0 or ). 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 The duodecimal system (also known as base 12, dozenal, or, rarely, uncial) is a positional notation numeral system using twelve as its base. The number twelve (that is, the number written as "12" in the decimal numerical system) is instead writ ...

(written with digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, , ): The natural Eisenstein primes are exactly the natural primes ending with 5 or (i.e. the natural primes congruent to ). (The natural primes that are prime in the Gaussian integers are exactly the natural primes ending with 7 or , i.e., the natural primes congruent to ).)

Examples

The first few Eisenstein primes that equal a natural prime are: : 2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ... . Natural primes that are congruent to 0 or 1 modulo 3 are ''not'' Eisenstein primes: they admit nontrivial factorizations in Z 'ω'' For example: : : . In general, if a natural prime ''p'' is 1 modulo 3 and can therefore be written as , then it factorizes over Z 'ω''as : . Some non-real Eisenstein primes are : , , , , , , . Up to conjugacy and unit multiples, the primes listed above, together with 2 and 5, are all the Eisenstein primes of

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...

not exceeding 7.

Large primes

, the largest known (real) Eisenstein prime is the ninth

largest known prime The largest known prime number () is , a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018. A prime number is a posi ...

, discovered by Péter Szabolcs and

PrimeGrid PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing ...

.Chris Caldwell,
The Top Twenty: Largest Known Primes
from The

Prime Pages The PrimePages is a website about prime numbers maintained by Chris Caldwell at the University of Tennessee at Martin. The site maintains the list of the "5,000 largest known primes", selected smaller primes of special forms, and many "top twenty" ...

. Retrieved 2019-09-18. All larger known primes are

Mersenne prime In mathematics, a Mersenne prime is a prime number that is one less than a power of two. That is, it is a prime number of the form for some integer . They are named after Marin Mersenne, a French Minim friar, who studied them in the early 17t ...

s, discovered by

GIMPS The Great Internet Mersenne Prime Search (GIMPS) is a collaborative project of volunteers who use freely available software to search for Mersenne prime numbers. GIMPS was founded in 1996 by George Woltman, who also wrote the Prime95 client and ...

. Real Eisenstein primes are congruent to , and all Mersenne primes greater than 3 are congruent to ; thus no Mersenne prime is an Eisenstein prime.

References

{{Prime number classes Classes of prime numbers Cyclotomic fields

Characterization

Examples

Large primes

See also

References