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| {{for|the unrelated concept of an Eisenstein prime of a modular curve|Eisenstein ideal}}
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| [[Image:EisensteinPrimes-01.svg|360px|right|thumb|Small Eisenstein primes. Those on the green axes are associate to a natural prime of the form 3''n'' − 1. All others have an absolute value squared equal to a natural prime.]]
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| In [[mathematics]], an '''Eisenstein prime''' is an [[Eisenstein integer]]
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| :<math>z = a + b\,\omega\qquad(\omega = e^{2\pi i/3})</math>
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| that is [[irreducible element|irreducible]] (or equivalently [[prime element|prime]]) in the ring-theoretic sense: its only Eisenstein [[divisor]]s are the [[unit (ring theory)|unit]]s (±1, ±ω, ±ω<sup>2</sup>), ''a'' + ''b''ω itself and its associates.
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| The associates (unit multiples) and the [[complex conjugate]] of any Eisenstein prime are also prime.
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| ==Characterization==
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| An Eisenstein integer ''z'' = ''a'' + ''b''ω is an Eisenstein prime if and only if either of the following (mutually exclusive) conditions hold:
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| #''z'' is equal to the product of a unit and a [[prime number|natural prime]] of the form 3''n'' − 1,
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| #|''z''|<sup>2</sup> = ''a''<sup>2</sup> − ''ab'' + ''b''<sup>2</sup> is a natural prime (necessarily congruent to 0 or 1 modulo 3).
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| It follows that the absolute value squared of every Eisenstein prime is a natural prime or the square of a natural prime.
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| ==Examples==
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| The first few Eisenstein primes that equal a natural prime 3''n'' − 1 are:
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| :[[2 (number)|2]], [[5 (number)|5]], [[11 (number)|11]], [[17 (number)|17]], [[23 (number)|23]], [[29 (number)|29]], [[41 (number)|41]], [[47 (number)|47]], [[53 (number)|53]], [[59 (number)|59]], [[71 (number)|71]], [[83 (number)|83]], [[89 (number)|89]], [[101 (number)|101]] {{OEIS|id=A003627}}.
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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:
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| :3 = −(1 + 2ω)<sup>2</sup>
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| :7 = (3 + ω)(2 − ω).
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| Some non-real Eisenstein primes are
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| :2 + ω, 3 + ω, 4 + ω, 5 + 2ω, 6 + ω, 7 + ω, 7 + 3ω.
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| 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.
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| ==Large primes==
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| {{As of|2010|3}}, the largest known (real) Eisenstein prime is 19249 × 2<sup>13018586</sup> + 1, which is the tenth [[largest known prime]], discovered by Konstantin Agafonov.<ref>Chris Caldwell, "[http://primes.utm.edu/top20/page.php?id=3 The Top Twenty: Largest Known Primes]" from The [[Prime Pages]]. Retrieved 2010-03-12.</ref> All larger known primes are [[Mersenne prime]]s, discovered by [[GIMPS]]. Real Eisenstein primes are congruent to 2 mod 3, and Mersenne primes (except the smallest, 3) are congruent to 1 mod 3; thus no Mersenne prime is an Eisenstein prime.
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| ==See also==
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| * [[Gaussian prime]]
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| ==References==
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| {{Reflist}}
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| {{Prime number classes}}
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| [[Category:Classes of prime numbers]]
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| [[Category:Cyclotomic fields]]
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