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a that is s are the Unit s 1, 1+ω, ω, -1, -1-ω, -ω, and ''a''ω + ''b'' itself and its unit multiples. Here ω is the complex cube Root Of Unity : The Eisenstein primes are precisely those Eisenstein integers α which fulfil one of the following conditions: #α is equal to the product of a unit and 1 - ω, #α is equal to the product of a unit and a natural prime 3''n'' - 1, #α can be multiplied by an Eisenstein integer such that the product is a natural prime 3''n'' + 1. 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 which are listed in . Some non-real Eisenstein primes are 2 + ω, 3 + ω, 4 + ω, 5 + 2ω, 6 + ω, 7 + ω, 7 + 3ω The Complex Conjugate of any Eisenstein prime is another; multiplying an Eisenstein prime by any of the units 1, 1+ω, ω, -1, -1-ω, -ω also gives an Eisenstein prime. 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. Eisenstein primes are named after the mathematician Ferdinand Eisenstein . 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. |
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