“Eisenstein prime”的意思、由来-开放百科全书

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Eisenstein prime

释义

Characterization
Examples
Large primes
See also
References

{{for|the unrelated concept of an Eisenstein prime of a modular curve|Eisenstein ideal}}

In mathematics, an Eisenstein prime is an Eisenstein integer

that is irreducible (or equivalently prime) in the ring-theoretic sense: its only Eisenstein divisors are the units {{math|{±1, ±ω, ±ω²}}}, {{math|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 {{math|z {{=}} a + bω}} is an Eisenstein prime if and only if either of the following (mutually exclusive) conditions hold:

{{math|z}} is equal to the product of a unit and a natural prime of the form {{math|3n − 1}},
{{math|{{abs|z}}² {{=}} a² − ab + b²}} is a natural prime (necessarily congruent to 0 or {{nowrap|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, the natural Eisenstein primes are exactly the natural primes end with 5 or 3 (i.e. the natural primes congruent to {{nowrap|2 mod 3}}), the natural Gaussian primes are exactly the natural primes end with 7 or 3 (i.e. the natural primes congruent to {{nowrap|3 mod 4}}).

Examples

The first few Eisenstein primes that equal a natural prime {{math|3n − 1}} are:

2, 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, ... {{OEIS|id=A003627}}.

Natural primes that are congruent to 0 or 1 modulo 3 are not Eisenstein primes: they admit nontrivial factorizations in Z[ω]. For example:

{{math|3 {{=}} −(1 + 2ω)²}}

{{math|7 {{=}} (3 + ω)(2 - ω)}}.

Some non-real Eisenstein primes are

{{math|2 + ω}}, {{math|3 + ω}}, {{math|4 + ω}}, {{math|5 + 2ω}}, {{math|6 + ω}}, {{math|7 + ω}}, {{math|7 + 3ω}}.

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|2017|3}}, the largest known (real) Eisenstein prime is the seventh largest known prime {{nowrap|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 {{nowrap|2 mod 3}}, and all Mersenne primes are congruent to 0 or {{nowrap|1 mod 3}}; thus no Mersenne prime is an Eisenstein prime.

References

1. ^Chris Caldwell, "The Top Twenty: Largest Known Primes" from The Prime Pages. Retrieved 2017-03-14.

2 : Classes of prime numbers|Cyclotomic fields

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Characterization

Examples

Large primes

See also

References