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, an Eisenstein prime is an

Eisenstein integer In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form :z = a + b\omega , where and are integers and :\omega = \f ...

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

that is

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(or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units , itself and its associates. The associates (unit multiples) and the complex conjugate 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 (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,

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, ... . 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 , discovered by Péter Szabolcs and PrimeGrid.Chris Caldwell,
The Top Twenty: Largest Known Primes
from The Prime Pages. Retrieved 2019-09-18. All larger known primes are Mersenne primes, discovered by GIMPS. 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