Talk:Mersenne prime
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I'm no mathematician, but shouldn't the definition of Mersenne prime include that it has to be one less than an natural power of two? Also, in the chart of Mersenne primes, shouln't the first entry be 2^0 - 1, which is 0?
No. 0 is not a prime number. It's divisible by every other integer--129.15.228.164 22:48, 3 September 2006 (UTC)
It is easy to see that 2p-1 is prime iff p is prime... I'm not sure its really easy at all... maybe I'll wait for the proof to come up on the Prime numberss page; but in the mean time, the sentence should be reworded for a regular guy (like me) to know whats going on. --cM
It is not hard. If p is not prime, then you can write p as qr. It is easy to see that 2^q-1 and 2^r-1 are then divisors of 2^p-1 and therefore 2^p-1 is not prime either. The reason is obvious if you write 2^p-1 in the binary form. It looks like the digit "1" repeated p times (1111...). Similarly, 2^q-1 is the number 1 repeated q times. For example, 2^15-1 is 111111111111111, and 2^3-1 is 111, and the latter is obviously a divisor of the former because the ratio is 1001001001001. See also my text at the bottom. --Lumidek 14:04, 6 Jun 2004 (UTC)
Ok, I'll reword it. In fact, "iff" is wrong. If 2p-1 is prime, then p must itself be prime.
--Axel
"Iff" being the abbreviation used in logic for "if and only if"?
Yes.
Generally, it seems to me bad form to ever use the words "clearly", "it is easy to see", and similar in math articles here. These are assuming a certain audience. Even if something is really, really clear, it's probably best not to use this. There will be someone learning it for the first time (we all learned "clear" things a first time) and "clearly" or "easy" may be off-putting. Revolver 04:57, 8 Jun 2004 (UTC)
2^31 - 1 = 2,147,483,647 !? Isnt that a prime? (see Two_billion_one_hundred_forty-seven_million_four_hundred_eighty-three_thousand_six_hundred_forty-seven, alas it will probably be deleted soon :b!) // Noone
- The article says it is prime... so I suppose it is, until I get a proof of the contrary. --FvdP 18:59, 26 Feb 2004 (UTC)
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- I can confirm that it is indeed prime, but you'll have to take my word for it. -- Dissident 04:19, 27 Feb 2004 (UTC)
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- I can confirm it, too. I opened Mathematica, typed PrimeQ(2^31-1) (well, with square brackets), and it answered True. --Lumidek 13:56, 6 Jun 2004 (UTC)
[edit] Mersenne number
What exactly is a "Mersenne number"? I've seen three different difinitions as of late. Obviously to be a candidate for a Mersenne prime a number must be "one less than a prime power of two", but isn't a "Mersenne number" simply "one less than a power of two"? --Pascal666 05:02, 6 Jun 2004 (UTC)
- Mathworld defines it as one less than a power of two, so that's probably right. Fredrik (talk) 13:41, 6 Jun 2004 (UTC)
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- Sorry, but you seem to misunderstand the word "prime". A Mersenne number may generally be any number 2^k-1, but it is not the same thing as Mersenne prime. A Mersenne prime must also be a prime, which means that its only divisors are the number itself and the number one. That's true for many small Mersenne numbers, but false for most large numbers. The smallest counterexample is 2^4-1 which equals 15=3 times 5 which is not prime. Well, it is not hard to prove that if 2^p-1 is prime, then p itself must be also prime - because otherwise if p=qr, then 2^q-1 and 2^r-1 are divisors of 2^p-1. However, the condition that the exponent p must be prime is not sufficient. The smallest counterexample is 11 which is prime, but 2^11-1 = 2047 = 23 times 89 is not. --Lumidek 14:04, 6 Jun 2004 (UTC)
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- iff 2^p-1, where p is prime, is prime, then the number is a Mersenne prime. For a Mersenne number to be a Mersenne prime, both p and 2^p-1 must be prime. - UtherSRG 18:58, 8 Jun 2004 (UTC)
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[edit] Mathematicians names
After some exhaustive googling, I managed to fix all the links to point to articles using each mathematician's full name. However, I was completely unable to find "R.E. Powers"'s full name. Does anyone happen to know it?
[edit] M13 - when found?
The text says 'before 1461'. The table says 1456. Which is it?
If M(n)=2^n-1, is there a sequence n,M(n),M(M(n)),...,M(M(...M(n)...)) (k is the length of the sequence) where all the numbers are prime? What are the values of n for different values of k? Is there such a sequence for all k?
[edit] revert of a suspicious change
I have reverted a suspicious edit [1] by 202.163.246.145 claiming that M32 was found on April 1 instead of February 19 and M33 on February 1 instead of January 10. The contributor did not offer any justification for the change and all sources I checked (e.g. [2]) confirm the original dates. So I believe the edit has been just a vandalism/test. --Mormegil 10:57, 16 July 2005 (UTC)
[edit] 43rd Mersenne prime
On December 16th, the 43rd known Mersenne prime was announced. If you can't live without a guess, it could be, among other choices, M 30402457, found by Curtis Cooper, but because you have no reason to believe my random generator, there is no reason for you to propagate the hypothesis. ;-) --Lumidek 04:38, 21 December 2005 (UTC)
[edit] 39th Mersenne prime definitely confirmed
According to the Gimps Status page, all exponents up to 15,300,000 have been Lucas-Lehmer tested at least once. This definitely confirms that M13,466,917 is the 39th Mersenne prime.
—Herbee 20:55, 4 January 2006 (UTC)
- Typically they wait until they have double-checked each exponent, which means running two LL tests. Which is why on the Gimps Status page it says "Countdown to proving M(13466917) is the 39th Mersenne Prime: 114". I'm going to revert your change. Qutezuce 21:04, 4 January 2006 (UTC)
Qutezuce, do you know the unit for the countdown: days ? exponents ? --FvdP 21:13, 5 January 2006 (UTC) (I guess exponents --FvdP 21:24, 5 January 2006 (UTC))
[edit] Recent changes regarding credit for M32
There is an IP address that is making some changes to the credit for M32. I believe this table is based on the database at primes.utm.edu, more specifically this page. That page credits Slowinski and Gage with M32. The IP address claims to be involved in this find, and said that they decided to credit only Harwell because it was his machine. However I believe primes.utm.edu uses a different criteria to give credit. For example the most recent of the Mersenne primes found by GIMPS was found on a computer owned by a university, but they gave credit to the people who administered the program to run the prime-checking software on hundreds of machines across the campus. So for now I've reverted the change, but I hope that the IP address returns and we can discuss this futher on this talk page. Qutezuce 20:31, 30 January 2006 (UTC)
Yes, This is my exact point; Slowinski and Gage had nothing to do with this find apart from supplying the code to me. I installed the code on the Cray-2 in a modified way, allowing to be run only when the system was completely idle. I found the M32 exponant and imediately passed it on to David and Paul for verification. Subsequently Harwell was given credit for the find as it was their machine. Perhaps George Woltman should be given credit for all of the subsequent finds !. 09:20 GMT 3rd Feb 2006
- You bring up some good points on how exactly the credit should be assigned. In the case of George Woltman getting credit, he is already indirectly given credit by way of GIMPS receiving credit. How about we simply explain that M32 was found by a computer at Harwell lab running code written by Slowinski and Gage? Qutezuce 09:52, 3 February 2006 (UTC)
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- I think we should keep things as short as possible here. The circumstances and details of each discovery should be covered at Great Internet Mersenne Prime Search. Fredrik Johansson - talk - contribs 14:31, 3 February 2006 (UTC)
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- We're discussing M32, which was not found by GIMPS. Qutezuce 17:41, 3 February 2006 (UTC)
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- Sorry; not reading carefully. Anyway, the articles Curtis Cooper and Steven Boone should be deleted. Fredrik Johansson - talk - contribs 17:46, 3 February 2006 (UTC)
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As a follow up to this discussion, http://primes.utm.edu/notes/756839.html has been created, and http://primes.utm.edu/mersenne/index.html has been updated with the credits - et al. 13:47 20th Feb 2006 (GMT)
- Thanks for clearing that up. I added a link to that description to the table. Qutezuce 22:26, 20 February 2006 (UTC)
[edit] Mersenne Prime naming
Could someone please explain, possibly in the article, just how the naming convention works for these things? I mean, where does the choice of the number in the subscript come from, as in the 13 in M13, for example? Narxysus 07:56, 7 May 2006 (UTC)
[edit] 44th Known Mersenne Prime Probably Found!!
"On September 4, 2006, a computer reported finding the 44th known Mersenne prime. Verification will begin shortly, probably taking a week or so to complete. If it is verified, this will be GIMPS' tenth prime!" http://www.mersenne.org/prime.htm
NevilleDNZ 23:34, 4 September 2006 (UTC)
[edit] Why?
Could someone please explain (and add to the article) what Mersenne primes are useful for / why they are of any interest. Is it just because of their extraordinary size? Can you do stuff (not just pure arithmetics) with Mersenne primes that you can't do with other primes?
- Check out the Mersenne Twister. NBS525 18:55, 15 September 2006 (UTC)
- The recently discovered prime is nearly 10 million digits long. It is the largest number about which anything nontrivial is known. That makes it very interesting. But it is also vastly larger than any number that could be used to count or measure anything in the universe. That second fact rules out many kinds of utility.
- Rbraunwa 01:22, 16 September 2006 (UTC)
[edit] Some editing issues
I decided not to act immediately due to my incompetence, and let the more knowledgable resolve my concerns.
1 - There are no links to Mersenne in this article. I presume he's a person with his own article? 2 - I got to Mersenne Prime via clicking a link to a Mersenne Number. Does Mersenne number derserve its own article? 3 - Shouldn't there be some mention somewhere of the fact that a mersenne number in binary is of the form 11111<...>111? Manning 06:25, 4 January 2007 (UTC)
- 1 - Mersenne does have a link to his own article in this article which says: "The numbers are named after 17th century French mathematician Marin Mersenne, who provided a list of Mersenne primes with exponents up to 257". 2 - I don't think Mersenne numbers should have their own article. They were named after Mersenne primes (unlike the usual where X primes are named after X numbers) and they are primarily called Mersenne numbers when discussing Mersenne primes. Some other things about Mersenne numbers can be mentioned in power of two which is currently linked in the first line. 3 - I agree. It's obvious to mathematicians but many readers may not notice it and several sources mention it. I have added: "The binary representation of 2n − 1 is n repetitions of the digit 1. For example, 25 − 1 = 11111 in binary." PrimeHunter 12:01, 4 January 2007 (UTC)
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- Brilliant. Thanks PrimeHunter :) Manning 22:34, 4 January 2007 (UTC)
[edit] Request for help on proof of "3)" in Theorems about Mersenne primes section
Would someone good at number theory like to rewrite my proof of 3) to get rid of all my group theory crutches and make it more leisurely and explanatory? Thanks,Rich 05:35, 21 February 2007 (UTC)
