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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
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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).
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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.
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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
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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?
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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?
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369:? Otherwise people will wonder whether there's some reason for writing all six out explicitly, which as far as I can see there isn't. --
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I'm searching for a special name for primes of the form 3n+1. BTW, congratulations on identifying the largest known
Eisenstein prime.
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239:: "The positive Eisenstein primes with zero imaginary part are precisely the ordinary primes that are congruent to 2 (mod 3)".
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A reader notes an issue with the image used in this article. I have uploaded an image which will be used in an explanation.--
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The image in the article has not quite perfect angles. Here's a short cut for creating a line with such exact angles.
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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I think it would be nice if this article listed a few
Eisenstein primes with imaginary parts.
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Precisely. Note that the sequence in OEIS referenced in the article is defined as
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4pq1injbok, I've noticed you wrote in the article, "All larger known primes are
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A counterexample: 3n-1=8 is congruent to 2 mod 3 but 8 is not prime.
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4pq1injbok, please show the largest known
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I don't understand what the diagram wants to tell us. —
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Thanks. Note that together with 3, primes of form 3
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