Talk:Eisenstein prime

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Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Context, introduction, explanation all needed. Geometry guy 21:56, 20 May 2007 (UTC)

Suppose 3n-1 = 27653.2^9167433+1, where n is some positive integer. Then, solving for n gives some rational or irrational number. A contradiction. Therefore, 27653.2^9167433+1 is not an Eisenstein prime.

4pq1injbok, please show the largest known Eisenstein prime. Giftlite 16:20, 11 September 2005 (UTC)

I maintain that 27653.2^9167433+1 is of form 3n-1 (i.e. is congruent to 2 mod 3). Working modulo 3, 27653 is congruent to 2, and 2^9167433 is also congruent to 2 since the exponent 9167433 is odd. Therefore the prime in question is congruent to 2.2+1 = 5 ≡ 2 (mod 3). 4pq1injbok 01:43, 12 September 2005 (UTC)

A counterexample: 3n-1=8 is congruent to 2 mod 3 but 8 is not prime. Giftlite 02:25, 12 September 2005 (UTC)

Yes, but that's a different objection. I showed only that 27653.2^9167433+1 has form 3n-1; the primality of this number has to be established independently. My reference for primality is [1]. 4pq1injbok 03:30, 12 September 2005 (UTC)

Are you saying Eisenstein primes belong in 2 mod 3 class and if 1. p is prime and 2. p also belongs in 2 mod 3 class then p is an Eisenstein prime? Giftlite 00:56, 13 September 2005 (UTC)

Precisely. Note that the sequence in OEIS referenced in the article is defined as Primes of form 3n-1. Also, from Weisstein's Mathworld: "The positive Eisenstein primes with zero imaginary part are precisely the ordinary primes that are congruent to 2 (mod 3)". 4pq1injbok 02:36, 13 September 2005 (UTC)

I think it would be nice if this article listed a few Eisenstein primes with imaginary parts. Anton Mravcek 21:21, 13 September 2005 (UTC)

I've added some. 4pq1injbok 00:23, 14 September 2005 (UTC)

4pq1injbok, I've noticed you wrote in the article, "All larger known primes are Mersenne primes and therefore congruent to 1 mod 3." Are you also saying Mersenne primes belong in 1 mod 3 class and if 1. p is prime and 2. p belongs in 1 mod 3 class then p is a Mersenne prime? Giftlite 03:23, 14 September 2005 (UTC)

No. It's only true that all Mersenne primes are 1 mod 3, not the other direction. I don't know a special name for the set of all primes 1 mod 3. It just happens that the largest couple primes known are Mersennes because numbers of that form are easy to test for primality, so some big searches exist (like GIMPS). 4pq1injbok 05:38, 15 September 2005 (UTC)

I'm searching for a special name for primes of the form 3n+1. BTW, congratulations on identifying the largest known Eisenstein prime. Giftlite 00:14, 16 September 2005 (UTC)

Thanks. Note that together with 3, primes of form 3n+1 are exactly the prime values taken by the quadratic form x²-xy+y², i.e. norms of Eisenstein integers. I think this means that in an integer-sided triangle with an angle of 2pi/3, if the side opposite this angle is prime it must be of the form 3n+1. So maybe there's a name for these primes similar to "Pythagorean prime"? 4pq1injbok 03:42, 17 September 2005 (UTC)