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 2011-09-19, 17:04 #1 siegert81   Dec 2010 2·37 Posts Gaussian-Mersenne & Eisenstein-Mersenne primes Hello! Are there any projects or plans to search for Gaussian-Mersenne & Eisenstein-Mersenne primes? I've noticed some Gaussian-Mersenne primes in the top 5000. It looks like they've found 38 of them. Apparently, no large Eisenstein-Mersenne primes have been discovered. Just curious. It seems like PrimeGrid would be a great platform for such a project...
2011-09-19, 17:32   #2
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

2×33×139 Posts

Quote:
 Originally Posted by siegert81 Hello! Are there any projects or plans to search for Gaussian-Mersenne & Eisenstein-Mersenne primes? I've noticed some Gaussian-Mersenne primes in the top 5000. It looks like they've found 38 of them. Apparently, no large Eisenstein-Mersenne primes have been discovered. Just curious. It seems like PrimeGrid would be a great platform for such a project...
I hope not. We already have too many projects.

 2011-09-19, 17:36 #3 kar_bon     Mar 2006 Germany 34×37 Posts See here for a current status of Gaussian-Mersenne primes search. P.S. Also https://mersenneforum.org/showthread.php?t=19655 and https://mersenneforum.org/showthread.php?t=19235
 2022-12-13, 14:08 #4 Andrew Usher   Dec 2022 197 Posts I don't see any status there, but I didn't page through the whole thread. There's now an Eisenstein Mersenne in the top 5000, though Caldwell refuses to recognise such a category to parallel the Gaussian Mersennes - it is instead listed as Phi(3, 31118781 + 1)/3 (generalized unique) which is puzzling as normally a number that size with a denominator can't be proved prime (and his list excludes PRPs). That it can be rewritten without one is not obvious from that. There are now 26 Eisenstein Mersennes and 41 Gaussian Mersennes; considering the search limits, that is very close indeed to the theoretical 2/3 ratio.
2022-12-14, 04:45   #5
sweety439

"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

2×7×263 Posts

Quote:
 Originally Posted by siegert81 Hello! Are there any projects or plans to search for Gaussian-Mersenne & Eisenstein-Mersenne primes? I've noticed some Gaussian-Mersenne primes in the top 5000. It looks like they've found 38 of them. Apparently, no large Eisenstein-Mersenne primes have been discovered. Just curious. It seems like PrimeGrid would be a great platform for such a project...
See https://mersenneforum.org/showthread.php?t=27552, Gaussian-Mersenne primes are primes of the form Phi((4*p)L,2) or Phi((4*p)M,2), where p is prime, including https://oeis.org/A007670 (L part, p == +-1 mod 8), https://oeis.org/A125742 (L part, p == +-3 mod 8), https://oeis.org/A124165 (M part, p == +-1 mod 8), https://oeis.org/A007671 (M part, p == +-3 mod 8)

2022-12-14, 07:19   #6
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

19×232 Posts

Quote:
 Originally Posted by Andrew Usher That it can be rewritten without one is not obvious from that. ...with a denominator can't be proved prime (and his list excludes PRPs)
Andrew Usher, if something is not obvious to you, just ask. Help is here! ( modesty aside, this is my prime after all :-) )

"You asked for it? You got it! Toyota!" (a forgotten slogan from '90s)
Anyway,
Phi(3,3^a+1)/3 = what is it?
Phi(3,x) = ๐ท3(x) = x^2+x+1, so ::
Phi(3,3^a+1)/3 = ((3^a+1)^2+3^a+1+1)/3 = (32a+2*3^a+1+3^a+1+1)/3 = 32a-1+3a+1,
and then you simply use the N-1 proof (because N-1 is nearly 50% fully factored, >> 33% factored).
Same for Gaussian-Mersennes.
More questions?

 2022-12-14, 13:49 #7 Andrew Usher   Dec 2022 19710 Posts I had already done that calculation for the general case, proving the the GMs and EMs are unique primes. They are also expressible easily as x^2 + y^2 and x^2 + xy + y^2 (respectively) with x and y consecutive, and therefore subsequences of OEIS A027862 and A002407, primes of those forms. How do you think I knew that was an EM, anyway? I was saying that, though both expressions are correct, Caldwell really should give the one used to discover and prove it rather than one that may cause even momentary confusion as to "what is this number doing here?".
2022-12-14, 16:22   #8
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

100111010000112 Posts

Quote:
 Originally Posted by Andrew Usher ... Caldwell really should give the one used to discover and prove it rather than one that may cause even momentary confusion as to "what is this number doing here?".
First of all, it is pointless to write on this forum what Caldwell should or should not do. Caldwell doesn't read this forum! You have something to tell him? You tell him! His email is on every page of his site.

Second, you give him surprisingly low credit in several ways:
a) he actually knows math (he is a full Math Professor! :) ) and if you ever tried to submit some nonsense 'prime' to his site, then you would know what he always does: first his server does computations, second - he contacts the submitter, third - he deletes if he dosn't get an answer.
b) Caldwell _does_ prove every submission, independently (except very few special ones, and then, only from users with math reputation, e.g. Woltman, Enge, ECPP people) - and this is _also_ on his webpages. Scroll the pages. What is there in them, in the middle?
c) have you actually read the webpage for this prime? It has a comment that lists the second representation of this prime. Your confusion is not something Caldwell can help, if you behave willfully ignorantly.

2022-12-14, 16:43   #9
Batalov

"Serge"
Mar 2008
Phi(4,2^7658614+1)/2

1005110 Posts

Quote:
 Originally Posted by kar_bon See here for a current status of Gaussian-Mersenne primes search. P.S. Also https://mersenneforum.org/showthread.php?t=19655 and https://mersenneforum.org/showthread.php?t=19235

 2022-12-15, 14:08 #10 Andrew Usher   Dec 2022 197 Posts I gave my opinion. Whether I consider it worth my time to try to change his mind is not even your business. You have not actually argued against said opinion and the fact that you posted the comment on the prime's page indicates that you might well agree with it. I don't read, much less remember, every page on Caldwell's site, so if you want to direct me to a specific point you need to point out what page it is on. I in fact didn't even recall that there were pages for individual primes. All your points are irrelevant to mine anyway, even if completely true, and the fact that the second representation was added by you yourself completely defeats that one. And your second post is quite obviously "automatic gainsaying". I won't reply to the silly insinuations, and I don't understand why you want to derail threads with your ignorant and uncivil - I'm not sure what noun applies, so readers may fill in the blank ;) Last fiddled with by Andrew Usher on 2022-12-15 at 14:08
 2022-12-15, 20:15 #11 paulunderwood     Sep 2002 Database er0rr 2×33×83 Posts Is this the relevant page: https://primes.utm.edu/top20/page.php?id=41 ?

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