Talk:Mersenne prime: Difference between revisions

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It can be easily seen that if p is an odd prime, then 2^p - 1 ≡ 7 or 31 (mod 40). It can be easily proven that 2^p - 1 ≡ 7 or 11 (mod 20). This is because 2^p - 1 ≡ 3 (mod 4) and 2^p - 1 ≡ 7 or 1 (mod 10). So, in the case that 2^p - 1 ≡ 7 (mod 10), 2^p - 1 ≡ 7 (mod 20) since for all integers k, 20k + 7 ≡ 3 (mod 4) and 20k + 17 ≡ 1 (mod 4) ≠ 3 (mod 4). In the case that 2^p - 1 ≡ 1 (mod 10), 2^p - 1 ≡ 11 (mod 20) since for all integers k, 20k + 11 ≡ 3 (mod 4) and 20k + 1 ≡ 1 (mod 4) ≠ 3 (mod 4). However, I need someone to go further and prove that 2^p - 1 ≠ 11 or 27 (mod 40). This is a necessary condition to prove that 2^p - 1 ≡ 7 or 31 (mod 40). PhiEaglesfan712 15:52, 13 July 2007 (UTC)[reply]

2⁴ ≡ 16 (mod 40) and 16² ≡ 16, so 16^k = 2^4k ≡ 16 for all k > 0. Thus 2^4k+1 − 1 ≡ 16·2 − 1 ≡ 31, and 2^4k+3 − 1 ≡ 16·8 − 1 ≡ 7 (mod 40). Notice that by a similar argument, because 16² − 16 = 240, it is possible to give the stronger result that for all k > 0, 2^4k+1 − 1 ≡ 31 (mod 240), and 2^4k+3 − 1 ≡ 127 (mod 240). John Blythe Dobson 06:44, 14 August 2007 (UTC)[reply]

From the above, it follows immediately that for a modulus which is any divisor of 240, the Mersenne numbers with k > 0 fall in at most two of the residue classes belonging to that modulus. Thus, for example, in decimal notation they always end in 1 or 7 (which happens to be true even when k = 0). John Blythe Dobson 01:45, 15 August 2007 (UTC)[reply]

It is in fact possible to give a slightly stronger result than the one I give above: for all k > 0, 2^4k+1 − 1 ≡ 31 (mod 480), and 2^4k+3 − 1 ≡ 127 (mod 480). This is because for k > 0, 2^4k+1 − 2⁵ = 2⁵(2^4k−4 − 1), and 2^4k+3 − 2⁷ = 2⁷(2^4k−4 − 1), and the factor (2^4k−4 − 1) = 16^k−1 − 1 which appears in each case is, by an identity appearing on the main page, divisible by 16 − 1 = 15. Note that regardless of whether p is of the form 4k+1 or 4k+3, these factorizations are divisible by 2⁵·15 = 480, which justifies the opening statement. Using arguments similar to these, it can be shown that all Mersenne numbers with p > 3 are ≡ 31 (mod 96), all Mersenne numbers with p > 3 are ≡ 31 or 127 (mod 288) according as p ≡ 5 or 1 (mod 6), etc., etc. John Blythe Dobson (talk) 04:19, 5 March 2009 (UTC)[reply]

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)[reply]

It is easy to see that 2^p-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 2^p-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)

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)

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)

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)

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)

I know what "prime" means and have not misunderstood anything. Before I corrected it, the article said that a Mersenne number is a prime power of two minus one (2^p-1, where p is prime), which is indeed wrong. Fredrik (talk) 07:49, 8 Jun 2004 (UTC)

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)

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?

The index to Joachim von Zur Gathen & Jürgen Gerhard, Modern Computer Algebra, 2nd ed., lists a Raymond Earnest Powers on pp. 559 and 744, but I can't view the actual pages in Google Books so cannot confirm that this is the right person (p. 559 falls in a chapter on the factorization of integers). His 1911 paper in the American Mathematical Monthly, in which he signs himself "R.E. Powers, Denver, Colorado," and in his 1914 note in the Proceedings of the London Mathematical Society which is credited simply to "R.E. Powers" seem to be his only known publications. I can't find corroboration of the name in Biography and Genealogy Master Index, WorldCat, Mathematical Reviews, JStor, or any other page found by Google, even using all obvious spelling variations. John Blythe Dobson 05:25, 14 August 2007 (UTC)[reply]

Hugh C. Williams, Édouard Lucas and Primality Testing (New York, etc.: Wiley & Sons, 1998), p. 153, refers to "R.E. Powers, an employee of the Denver and Rio Grande Western Railroad Company of Denver, Colorado," but does not supply his full name. Williams' extensive bibliography shows two further articles beyond those I previously mentioned. Powers, whose first known publication was the 1911 article cited above and whose last known article appeared in 1934, was still alive on 7 September 1935, when he wrote a letter cited in Williams, p. 504. John Blythe Dobson 21:05, 9 October 2007 (UTC)[reply]

The text says 'before 1461'. The table says 1456. Which is it?

MathWorld says 1461. (Anyway, 1456 is 'before 1461'. :-) ) --Mormegil 14:38, 2 Mar 2005 (UTC)

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?

One source I read had Codice Palatino as the discoverer of M13 in 573. Another source had Cataldi as the first to prove the primality in 1588. The table has 1456 as the date of discovery of M13... but the table has no discoverer. This is why I believe that the table is the least credible of the 3... The other two have not only a date of discovery, but also a discoverer.

Either your source is wrong or you have misunderstood it. "Codice Palatino 573" is not a person and year but a title and number given to a codex. See [1] which says "We will show that the 5th perfect number was calculated in 1458 by the author of the Codice Palatino 573 of the Biblioteca Nazionale of Florence". If your source is [2] then it says "8191=2¹³-1 (before 1458, Codice Palatino 573)" which means that Codice Palatino 573 is the source to the year 1458. PrimeHunter 21:43, 14 June 2007 (UTC)[reply]

I have reverted a suspicious edit [3] 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. [4]) confirm the original dates. So I believe the edit has been just a vandalism/test. --Mormegil 10:57, 16 July 2005 (UTC)[reply]

Another source, [5], found by PrimeHunter, reveals that M33 was actually found on January 4, 1994. PhiEaglesfan712 14:42, 13 August 2007 (UTC)[reply]

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)[reply]

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 M_13,466,917 is the 39th Mersenne prime.
—Herbee 20:55, 4 January 2006 (UTC)[reply]

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)[reply]

I stand corrected.<blush/>
—Herbee 21:26, 4 January 2006 (UTC)[reply]

You're not the only one to blush from the same mistake ;-) (I did it) --FvdP 21:13, 5 January 2006 (UTC)[reply]

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))[reply]

Your guess would be correct. To quote the site http://www.mersenne.org/status.htm:

July 10, 2006: Double-checking proves M(13466917) is the 39th Mersenne prime.

April 23, 2007: All exponents below 15,000,000 double-checked.

From the foregoing it appears that exponents are being double-checked at the rate of about a million every six months.

So M(20996011) may not be confirmed as the 40th until Spring 2010 unless exponents are double-checked more quickly.

Glenn L 09:23, 2 July 2007 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

We're discussing M32, which was not found by GIMPS. Qutezuce 17:41, 3 February 2006 (UTC)[reply]

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)[reply]

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)[reply]

The correct notation is "Mersenne 32", not M32... In fact, M32 is not even prime because 32 is composite. —Preceding unsigned comment added by PhiEaglesfan712 (talk • contribs)

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)[reply]

The very beginning of the main article defines the convention:

M_n = 2ⁿ − 1.

However, M₁₃ should not be confused with M13:

M₁₃ or 2¹³ - 1 (8,191) is M5, the 5th Mersenne prime.

M13, the 13th Mersenne prime, is M₅₂₁ or 2⁵²¹ - 1 (≈ 6.86479766 × 10¹⁵⁶).

Glenn L 06:36, 9 July 2007 (UTC)[reply]

You guys are still using the notation the wrong way...

M13 = 2¹³ - 1 = 8191 is "Mersenne 5" not M5

M5 = 2⁵ - 1 = 31, which in fact is "Mersenne 3"

Mersenne 13 is M521 = 2⁵²¹ - 1 (≈ 6.8648 × 10¹⁵⁶). —The preceding unsigned comment was added by PhiEaglesfan712 (talk • contribs).

Different notations are used by different sources (although I have never seen your "Mersenne 13"), and the article explains the notation it uses. I don't see a problem. PrimeHunter 23:39, 9 July 2007 (UTC)[reply]

Here is an article that uses my "Mersenne n" notation - http://primes.utm.edu/top20/page.php?id=3 PhiEaglesfan712 16:08, 10 July 2007 (UTC)[reply]

That is a site with prime tables where "prime" is omitted from table names for convenience, because they are all prime. For example, http://primes.utm.edu/top20/page.php?id=30 at the same site says "factorial" about a factorial prime even though it is not a factorial. PrimeHunter 17:19, 10 July 2007 (UTC)[reply]

"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)[reply]

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)[reply]

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)[reply]

If we denote the recently discovered Mersenne prime by p, then the perfect number associated with it is approximately ${\displaystyle (p^{2})/2}$, which is bigger. We know about as much about this perfect number as we do about p. 10^(10^(10^(10^(10)))) is a still bigger number about which we know something,including its divisors.67.169.227.132 04:39, 8 April 2007 (UTC)Richard L. Peterson, forgot to log in.[reply]

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)[reply]

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)[reply]

Brilliant. Thanks PrimeHunter :) Manning 22:34, 4 January 2007 (UTC)[reply]

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)[reply]

I wouldn't necessarily remove the group-theoretic proof, as it is perfectly valid (of course) and contains useful cross-references to other Wikipedia articles. However, I agree that it would be preferable to have a self-contained proof which relies only on concepts from elementary number theory. What follows may not be the most elegant proof possible, but it is probably the easiest for a general audience to understand. If you agree, please feel free to use it; I didn't want to change your work. Here goes: If q divides 2^p − 1 then 2^p ≡ 1 (mod q). By Fermat's Little Theorem, 2^(q − 1) ≡ 1 (mod q). Assume there exists such a p which does not divide q − 1. Then as p and q − 1 must be relatively prime, a similar application of Fermat's Little Theorem says that (q − 1)^(p − 1) ≡ 1 (mod p). Thus there is a number x ≡ (q − 1)^(p − 2) for which (q − 1)·x ≡ 1 (mod p), and therefore a number k for which (q − 1)·x − 1 = kp. Since 2^(q − 1) ≡ 1 (mod q), raising both sides of the congruence to the power x gives 2^(q − 1)x ≡ 1, and since 2^p ≡ 1 (mod q), raising both sides of the congruence to the power k gives 2^kp ≡ 1. Thus 2^(q − 1)x ÷ 2^kp = 2^{(q − 1)x − kp} ≡ 1 (mod q). But by definition, (q − 1)x − kp = 1, implying that 2¹ ≡ 1 (mod q); in other words, that q divides 1. Thus the initial assumption that p does not divide q − 1 is untenable. John Blythe Dobson 02:44, 22 September 2007 (UTC)[reply]

put yours in just now.Rich (talk) 03:46, 10 June 2008 (UTC)[reply]

Thanks, Rich. I didn't check this page for a long time, and just got your message today.John Blythe Dobson (talk) 04:00, 18 February 2009 (UTC)[reply]

I saw the call for a citation following the statement of how many pages would be required to print M44. So I fired up Microsoft Word 2003, imported the text of M44 (downloaded from www.mersenne.org), set the margins to one inch and the typeface to 12-point Times New Roman, and voila: two thousand and seven hundred and thirty and four pages (to be precise, 2,733 full pages each containing 46 lines of 78 digits, plus a partial page containing 30 full lines and a partial line containing 14 digits). This result differs slightly from the number of pages indicated by Jmalc, which I attribute to slight differences in line spacing between Jmalc's word processor and mine.

Unfortunately, this constitutes original research and thus is not usable here. My chances of getting this fact published in a reputable journal is small (at best), so there will be no citation. Too bad. Jmalc's result is also original research. Perhaps, under the new policies, this whole statement should be purged. — SWWright^Talk 20:12, 16 April 2007 (UTC)[reply]

I think it would be OK if the article dropped the font and margin specifications and instead said something trivially verifiable with a calculator, for example like: If a page has x lines with y digits then z pages would be needed. Some reasonable round numbers would be 50 lines with 80 digits. PrimeHunter 22:46, 16 April 2007 (UTC)[reply]

80 characters a line is sort of a defacto standard in the computer science world, and 50 lines is two screens worth of text on a standard terminal. Rounding up that's 2453 pages or 4905 screens of text on a terminal. Or for another statistic: the ascii text representation of the digits takes up 9.35MB of storage.

Not sure if this section should even be part of the article in the first place but it's not hard to see the current info is wrong since by the most significant digits in the factors (5 and 2), the most significant digit in the product should thus be a 1 instead of a 5. Not sure if the value or the factors are wrong....can someone verify? Sentri 01:42, 23 May 2007 (UTC)[reply]

Oh nvm...my bad...didn't see the first factor Sentri 01:43, 23 May 2007 (UTC)[reply]

I think codice is the plural of codex. Kope 05:21, 14 June 2007 (UTC)[reply]

Actually, codice is the Italian form of the word codex. The Latin plural is codices. John Blythe Dobson 05:30, 14 August 2007 (UTC)[reply]

This isn't true if one of x or y is zero, or if x=y. However, outside of those degenerate cases, I'm not entirely sure that x^a-y^a|x^b-y^b implies a|b (of course, the proof of the converse is straightforward). 192.236.44.130 07:30, 12 August 2007 (UTC)[reply]

PhiEaglesfan712 has changed the definition of Mersenne number and posted in an old section. See section 2 #Mersenne number for discussion. PrimeHunter 02:08, 15 August 2007 (UTC)[reply]

I have moved the discussion here. Arcfrk 02:17, 20 August 2007 (UTC)[reply]

A "Mersenne number" is one less than a prime-numbered power of two. [6] If we are not going to count 2 as a Fermat prime, (since it can be written 2^0 + 1), then it does not make any sense to me to define a "Mersenne number" as simply "one less than a power of two". It makes much more sense to me to define a Mersenne number as "one less than a prime-numbered power of two." —The preceding unsigned comment was added by PhiEaglesfan712 (talk • contribs).

I strongly suggest you discuss on talk pages before changing definitions of article subjects that require significant changes to the article text and other articles, as you did in Mersenne number [7] and Double Mersenne number [8]. And using edit summary "fix definition of Double Mersenne number - with source" [9] followed by deletion of a link to a reliable and popular source that uses the former definition [10] is misleading. In addition, it's unclear whether your source here [11] actually requires prime exponent in Mersenne numbers. The glossary at the same site, http://primes.utm.edu/glossary/page.php?sort=MersenneNumber, does not require prime exponent but just says that many authors do.

Fermat numbers are of form 2^(2^n)+1 and Fermat primes are prime Fermat numbers. 2 = 2^0+1 cannot be written as 2^(2^n)+1, so it's not a Fermat number and therefore not a Fermat prime. I don't see any reason why this should affect the definition of Mersenne numbers. And Wikipedia definitions should be based on reliable sources and not on what "makes sense" to an editor making original research analogies. Reliable sources use two different definitions so we should say both are in use and not delete one of them like you did. For practical reasons we either have to choose one of them in our articles, or write two versions of many things which would be an unnecessary mess. I recommend we choose 2^n-1 for integer n and not restricted to prime n. It's common, for example in http://mathworld.wolfram.com/MersenneNumber.html. It's practical to use the more general definition when writing things that are relevant for both versions. Other articles, for example Repunit#Generalizations, would require changes if we change to PhiEaglesfan712's definition (and Mersenne number would require further changes). Many things depend on a common definition we have used for a long time and we should not change it without very good reason. PrimeHunter 23:46, 14 August 2007 (UTC)[reply]

Well, I am sorry for changing the definitions without first thoroughly discussing on the talk pages. However, I was taught that the definition of a Mersenne number is a number that is one less than a prime-numbered power of two. I was first taught this definition in Autumn 2000, when I was only 12 years old. In fact, prior to 2007, I never heard of the alternative definition of a Mersenne number, which is simply one less than a power of two. By the way, my source [12] does require a Mersenne number to have a prime exponent. If you read the question "Is every Mersenne number square-free?" in the Conjectures and Unsolved Problems page, then it can be deduced that the author requires a Mersenne number to have a prime exponent. (This is simple because by taking the non-prime exponent6, then 2⁶ - 1 = 63 = 7 × 9 = 7 × 3² and is not square-free.) I absolutely agree with you that many things depend on a common definition we have used for a very long time and we should not change it without a very good reason. However, the definition that I have used for the past 7 years that a Mersenne number is a number that is one less than a prime-numbered power of two is the one I have accepted. And, I have no reason to conform to the other definition without a very good reason. PhiEaglesfan712 16:01, 15 August 2007 (UTC)[reply]

Your source doesn't have an actual definition and seems unsuited as reference for a mathematical definition. That original research can indicate which definition it may have in mind is insufficient (and as mentioned the same site doesn't require prime exponent in the glossary). Wikipedia is a collaborative project. Try putting your ego aside. "I have no reason to conform to the other definition" is in my opinion a very inappropriate reason to suddenly make undiscussed changes of a common definition which has been used for years by many editors in many articles, and is used by many reliable sources, for example MathWorld [13] and American Scientist [14]. If you haven't seen that definition before 2007 then you have probably only read little about prime numbers before 2007, but that's no surprise if you are 19. Your redefinition has created many inconsistencies both internally in Mersenne number and in other articles linking to it. Cleaning up this mess would be non-trivial and reverting your edits would be trivial. I suggest the latter but would like to first hear input from other editors. If we keep PhiEaglesfan712's definition then we should at least say that the other definition also exists (like we did before PhiEaglesfan712). Otherwise the article will be confusing and look wrong to those readers who are used to the other definition, and it might also be problematic with respect to Wikipedia:Neutral point of view (although that policy may not apply well to mathematical definitions). PrimeHunter 23:04, 15 August 2007 (UTC)[reply]

The primepages link you cite doesn't require the exponent of a Mersenne number to be prime. The particular question on that page, though, is whether Mersenne numbers with prime exponents are all squarefree. CRGreathouse (t | c) 16:22, 20 August 2007 (UTC)[reply]

Paulo Ribenboim, The Book of Prime Number Records, 2nd edition, Chapter 2, Section VII Mersenne Numbers, says 'The numbers ${\displaystyle M_{q}=2^{q}-1}$ (with q prime) are called Mersenne numbers'. DRLB 17:19, 20 August 2007 (UTC)[reply]

For what it's worth, Sloane's encyclopedia agrees -- it calls the numbers with prime exponents Mersenne numbers. Granted, it notes that some don't require the exponent to be prime (in sequence A000225). What other good references do we have? I'm going to check my Crandall and Pomerance when I get home. CRGreathouse (t | c) 14:49, 21 August 2007 (UTC)[reply]

I have rewritten the lead, trying to make it less terse and more readable. I have also restored the old definition of the Mersenne number. One of the most authoritative sources on the subject, Crandall and Pomerance, define the "Mersenne numbers" to be the numbers of the form

${\displaystyle M_{q}=2^{q}-1.}$

They are using a diffirent letter as a mnemonic devise to indicate that we are mostly interested in these numbers for certain special values of q, but the definition is worded for arbitrary natural exponent q. A footnote with an explantion of conventions may be added, but I felt reluctant to put it into the lead. Alternatively, an explanation can be given later in the main text. PrimeHunter gave above a compelling consistency argument for keeping the present convention.

Other changes to the lead: streamlined typography (hopefully, correct and consistent); removed clutter; added a brief mention of Lucas–Lehmer test; moved repunit property closer to the definition of Mersenne numbers, in order not to break the continuity of the exposition of Mersenne primes. Arcfrk 02:26, 20 August 2007 (UTC)[reply]

Thanks for your good work. I have made some small changes. I think Double Mersenne number should also be changed back to allow composite exponents, by reverting to the version by Giftlite (the See also link to Fermat number can stay). PrimeHunter 02:50, 21 August 2007 (UTC)[reply]

At the bottom of the List of Known Mersenne Primes section, a statistic is given that it would take 2769 pages to display M₄₄ in a standard word processor, and then it's marked as "Citation Needed". Unfortunately, an exact figure (and by extension, the validity of any citation) may be impossible, as this number will vary a little bit, since some word processors and some printers will alter the space between characters or lines ever so slightly (as well as several other less-critical factors).

However, since we know the number of digits in M₄₄, and numbers are usually fixed-width, even in proportionally-spaced fonts, it's easy enough to attempt to duplicate the statistic with a simple macro. Using Microsoft Word XP for the PC, I got 2734 pages using all 0's. This certainly seems to be in line with the original poster's comment, though not exactly the same for the various reasons just noted. --Rob 22:17, 15 September 2007 (UTC)[reply]

It's also discussed higher up in #A bit of original research. Great Internet Mersenne Prime Search says:

The number M_32582657 has 9,808,358 digits. To help visualize the size of this number, a standard word processor layout (50 lines per page, 75 digits per line) would require 2,616 pages to display it.

I think it's best to be specific about number of lines per page and digits per line, and use the same numbers in both articles. PrimeHunter 00:24, 16 September 2007 (UTC)[reply]

I have done that in [15] after somebody removed the fact tag without adding a source. PrimeHunter (talk) 00:42, 30 January 2008 (UTC)[reply]

This strikes me as a tad pompous:

Perhaps even more embarrassingly, it is not known whether infinitely many Mersenne numbers with prime exponents are composite

Could this be reworded to avoid the implication that Wikipedia is ashamed of mathematicians who can't prove a property so painfully simple, when most of humanity can't even understand what that property is? --P3d0 02:51, 22 September 2007 (UTC)[reply]

This states that xa − ya | xb − yb if and only if a|b —Preceding unsigned comment added by Barneypitt (talk • contribs) 22:57, 18 January 2008 (UTC)[reply]

Thanks all for fascinating article, I would appreciate a x-reference next to the statement...

   This states that xa − ya | xb − yb if and only if a|b

under the "Searching for Mersenne primes" section. If anyone would like to let me know how I add a "reference required" comment instead of using the chat page then thanks (sorry, newbie).

Thanks again, B —Preceding unsigned comment added by Barneypitt (talk • contribs) 23:02, 18 January 2008 (UTC)[reply]

Shouldn't the article explain who Mersenne was? A bit lower down, the article says that Euclid worked on them, so how did the Mersenne number/prime come to be named after Mersenne? AnteaterZot (talk) 12:00, 14 March 2008 (UTC)[reply]

What do you mean? Mersenne prime#History links to Marin Mersenne, and the section does say what Mersenne did: He made a claim about which numbers up to M₂₅₇ are prime. That is apparently all he did and some of his numbers were wrong, but the claim is famous. PrimeHunter (talk) 00:29, 15 March 2008 (UTC)[reply]

I was reading a book in which Mersenne Numbers and prime numbers where referred in a way it made me confuse, so I came to Wikipedia for clarify my doubts, only to find that this article confused me too. It was only after re-reading and checking the discussion page that I finally concluded that not all Mersenne numbers are prime numbers too.

So, I ask: Why not making an exclusive article for Mersenne numbers instead of redirecting it to the article of Mersenne primes?

--Francisco Albani (talk) 00:54, 24 March 2008 (UTC)[reply]

The term Mersenne number is mainly used in the context of discussing which of them are Mersenne primes so I think it's good to have them in the same article. In most other cases where "X prime" is defined as a "X number" that is prime, X number has independent notability and X primes were often studied much later and much less. PrimeHunter (talk) 01:46, 24 March 2008 (UTC)[reply]

I have removed a new section called "Distribution of Mersenne primes" with this text:

In 1992 Prof. Haizhong Zhou conjectured that if ${\displaystyle 2^{2^{n}}<p<2^{2^{n+1}}}$, then there are ${\displaystyle 2^{n+1}-1}$ Mersenne prime(s). Based on his conjecture, he presented a corollary: if ${\displaystyle p<2^{2^{n+1}}}$, then there are ${\displaystyle 2^{n+2}-n-2}$ Mersenne primes (${\displaystyle p}$ is prime number; ${\displaystyle n}$ is natural number)

p is the exponent in a Mersenne prime. The conjecture appears non-notable and I see no good reason to believe it. The conjecture was clearly made to fit 4 known data points: There is 1 Mersenne prime exponent p with 2^1 < p < 2^2, there are 3 with 2^2 < p < 2^4, there are 7 with 2^4 < p < 2^8, and there are 15 with 2^8 < p < 2^16. The conjecture then predicts 31 with 2^16 < p < 2^32. This may take decades to test. I guess the conjecture was made after experimenting with a lot of guesses involving different sets of data points. Given the law of small numbers, there is likely to be a small data set which matches some pattern by "coincidence", without following the pattern forever. If a good independent source is found then the conjecture might be mentioned in Mersenne conjectures. PrimeHunter (talk) 13:19, 27 April 2008 (UTC)[reply]

play in the article. Although MathWorld and Wikipedia define them as all numbers of form (2^n - 1), Sloanes Handbook of integer sequences leans towards requiring n to be prime. A couple sentences about what number theory texts prefer would be good.Rich (talk) 09:43, 13 May 2008 (UTC)[reply]

The article currently says "Mersenne gave no indication how he came up with his list..." This contradicts Dickson, History of the Theory of Numbers, I, 13 and note 61, who lists the criteria by which Mersenne selected his numbers. In 1647 Mersenne stated without proof that Mp is prime when p is a prime of one of the three forms 4^m + 1, 4^m + 3, or 2^m - 1 (the last, namely itself a Mersenne prime, being somewhat obscurely expressed by Mersenne). Applying this test to all Mersenne numbers below M8191 yields exactly Mersenne's four picks, namely M31, M67, M127, and M257.

So in the range Mersenne considered, his first two rules, yielding M67 and M257, scored 0% while his recursive rule, that Mp is a Mersenne prime when p is, yielding M31 and M127, scored 100%. Though the first two rules served him poorly, thanks to the third rule he ended up with a better overall score than he could have expected had he simply picked four primes at random between 19 and 257.

So while it's true that Mersenne gave no indication how he came up with his rules, that's not to say that he gave no indication how he came up with his list. If there are no objections by the end of the month I suggest adjusting the article accordingly. --Vaughan Pratt (talk) 04:16, 14 May 2008 (UTC)[reply]

This is also online at [16]. Apparently Mersenne published the alleged primes in 1644 without rules and then gave rules without arguments for the rules in 1647. PrimeHunter (talk) 10:43, 14 May 2008 (UTC)[reply]

Someone needs to make the above change in the text. Additionally, if Mersenne published M31 and M127 in 1647, why does the table attribute their discoveries to others a century or two later (M31 by Euler in 1772 and M127 by Lucas in 1876). Is Mersenne not given credit because his was only a conjecture and not a proof? If so, that should be explicitely stated somewhere. 68.73.93.0 (talk) 07:32, 27 September 2008 (UTC)[reply]

Yes he's not given credit because he didn't prove it. Nico92400 (talk) 08:03, 29 September 2008 (UTC)[reply]

I understand that a "Mersenne number is a number that is one less than a power of two". However, why do these numbers warrant a special name? what is their significance? --Sreifa01 (talk) 12:28, 26 May 2008 (UTC)[reply]

Integers one less than a power of two are important for computers because 2ⁿ-1 is the largest number which can be represented with a n-bit unsigned integer (or with a (n+1)-bit signed integer). This has lead to many limits like that on drives, files, memory, data sets, computations, ... (see also power of two). However, the term "Mersenne number" is mostly used when discussing which of them are Mersenne primes. That search goes back far longer than binary computers. The largest known prime has usually been a Mersenne prime and they get more computer time than other prime forms. Great Internet Mersenne Prime Search has tens of thousands of computers contributing around 29 teraflops. I guess this is caused by a combination of historical interest, a simple "beautiful" form, and the fast Lucas–Lehmer primality test (certain other forms have comparable speed today). PrimeHunter (talk) 13:12, 26 May 2008 (UTC)[reply]

(copied from talk:Prime_number) There's little more besides a headline stating more information to come soon, but http://mersenne.org/prime.htm claims to possibly have found the 45th mersenne prime number. Slashdot has also covered it. --76.85.144.126 (talk) 00:32, 28 August 2008 (UTC)[reply]

Yes, it has been at the end of Mersenne prime#List of known Mersenne primes for 4 days.[17] I don't think we should write more now if the prime is still unverified and of unknown size. Wikipedia is not a news site and doesn't have to bring the latest unreliable rumours about the value. PrimeHunter (talk) 00:54, 28 August 2008 (UTC)[reply]

For your info, the exponent probably won't be revealed until September 12. Georgia guy (talk) 21:49, 28 August 2008 (UTC)[reply]

This is not a place for rumors so no one is obliged to tell you that the exponent will be either 42801739 or 42760397, at any rate, more than 10 million digits. ;-) --Lumidek (talk) 08:11, 3 September 2008 (UTC)[reply]

On Saturday, GIMPS found one more, 46th Mersenne prime. ;-) [18] The exponent is probably around 40 million, too. It will take two weeks or so to check it. --Lumidek (talk) 07:08, 8 September 2008 (UTC)[reply]

It was added to the article yesterday.[19] PrimeHunter (talk) 10:48, 8 September 2008 (UTC)[reply]

Peoples, peoples, peoples - stop fighting over 44 in August versus 46 in September. The two new ones are to be announced within about 3 days. Until then, just leave it at 44 in August. Bubba73 (talk), 05:53, 15 September 2008 (UTC)[reply]

I agree and have removed it again. If two numbers are only allegedly known by a few people who keep them secret, then they shouldn't be called "known" without reservation. "known" implies publicly known. If the people are trusted then it could be said that it is known there are at least 46 Mersenne primes, but "46 Mersenne primes were known" is bad. PrimeHunter (talk) 12:41, 15 September 2008 (UTC)[reply]

Dear prime busters, I have updated the list of the Mersenne primes, up to M45,46. It should be OK, including the 9+9 digits, authors of the discovery, the number of decimal digits etc. but you are invited to recheck because I have used some tricks to get the digits. Please confirm it here that you got the same result. I didn't round the first 9 digits of M45 up - otherwise it would be 06 instead of 05 or something like that. The temporary footnotes for M45,46 were erased. Best wishes, Lubos Motl --Lumidek (talk) 13:16, 16 September 2008 (UTC)[reply]

The table contained

202254405...

316470267...

but I get

202254406890977335534...

316470269330255923143...

with mpmath (also verified with Mathematica). I have updated the table accordingly. Fredrik Johansson 14:51, 16 September 2008 (UTC)[reply]

I agree with your calculations, using Maple. Bubba73 (talk), 15:35, 16 September 2008 (UTC)[reply]

I don't see how the fact that a Mersenne prime is a base-2 repunit prime is a generalization. It is the same thing, right? (unless you speak of repunit primes in different bases.) Bubba73 (talk), 16:50, 16 September 2008 (UTC)[reply]

Well, I guess it is OK. It wasn't clear to me the first time I read it. Bubba73 (talk), 17:03, 16 September 2008 (UTC)[reply]

It is absolutely silly to try to write out a number name of a number with millions of digits as "X hundred Xty X gazillion, X hundred Xty X bajillion, X hundred Xty X zillion, X hundred Xty X jillion..." It can easily be impractical, and you can easily be forced to lose track. The best way to name huge numbers is just to pronounce each individual digit. Georgia guy (talk) 17:12, 16 September 2008 (UTC)[reply]

I disagree. When I see a large pile of digits one thing that comes to my mind is how would you say it. I have a poster of the first million digit mersenne in my office. A common remark I often hear is how would you even say that number. There is interest in the name of the number. It is an interesting fact that should be preserved (as it was with the 44th mersenne). It should be put back. —Preceding unsigned comment added by 32.155.100.156 (talk) 18:27, 16 September 2008 (UTC)[reply]

I agree with the edit to put the sentence back. I will add to the remark above that the naming system in question if far from silly. The name system referenced was co-invented by the mathematicians John Horton Conway and Landon Curt Noll. The latter found two Mersenne primes a few decades back. We should keep it as an interesting visilation fact: or change it to how high the stack of paper would be if it were printed on standard office laser printer paper. —Preceding unsigned comment added by 32.157.92.185 (talk) 19:50, 16 September 2008 (UTC)[reply]

Someone told me there was a bit of controversy over my edit. I didn't intend to put someone in a huff, however. Being the person who added the original text, I concur with the others who think it belongs.

I like the above comment about a paper stack. Since it is being added to a remark about helping visualize the size of the prime, a paper stack height might do the trick. 4321583 lines at 50 lines per page, double sided yields 43218 pages. Using standard 20lb office paper, that would require 86.436 reams (of 500 sheets). A 20lb ream is about 2 inches thick, so the name of the number would stand 172.872 inches or about 14 feet 5 inches or 4.39 meters high. That might be a better visualization than a page count. —Preceding unsigned comment added by Landon Curt Noll (talk • contribs) 21:21, 16 September 2008 (UTC)[reply]

1. Is it known for sure that he was 4th century, not 3rd?
2. Is there any reason to suppose he was the 1st to prove the result on perfect no.s?

Peter jackson (talk) 10:52, 19 September 2008 (UTC)[reply]

Just to let you know m31 has been syated wrongly in the list of primes. It reads 2147483647 and should be 2147483646 —Preceding unsigned comment added by 80.4.85.154 (talk) 15:43, 28 October 2008 (UTC)[reply]

Huh? M₃₁ is a prime and your value is even! The article is right: M₃₁ = 2³¹-1 = 2147483647. I guess you misunderstood something and subtracted 1 twice from 2³¹. PrimeHunter (talk) 17:25, 28 October 2008 (UTC)[reply]

I must be missing something here but 2³¹ = 2147483647 and 2³¹ -1 = 2147483646. Or is that not correct? —Preceding unsigned comment added by 80.4.85.154 (talk) 18:10, 28 October 2008 (UTC)[reply]

Ah my mistake -sorry! My calculator has a rounding error on it! I've just tried it on Excel and of course the answer is correct. I should have realised as the answer I got was an even number. Thanks once again! —Preceding unsigned comment added by 80.4.85.154 (talk) 18:18, 28 October 2008 (UTC)[reply]

The article states that the largest prime number known has "almost always been a Mersenne prime" and the foot note states that it has been this way since 1952, except for a small time range. How is since 1952 "almost always"? Asmeurer (talk ♬ contribs) 06:12, 20 November 2008 (UTC)[reply]

The article states that 2¹⁰³⁹ − 1 is the largest Mersenne number factorized. This does not make sense, as 2^43,112,609 − 1 (with factorization 2^43,112,609 − 1 = 2^43,112,609 − 1) is much, much larger. — Emil J. 14:44, 12 December 2008 (UTC)[reply]

Perhaps it should say that 2¹⁰³⁹ − 1 is the largest non-prime Mersenne number that has been factorized.—GraemeMcRae^talk 15:24, 12 December 2008 (UTC)[reply]

Perhaps, or perhaps not, do we really know that this is the correct interpretation? Is there a reliable source saying that no larger composite Mersenne has been factorized? — Emil J. 15:51, 12 December 2008 (UTC)[reply]

You're quite right to question this interpretation, I now know. I checked GIMPS [20], which reports, for example, that exponent 1747 was factored, the factors being 16811052664235873, 22831737099099330859999.—GraemeMcRae^talk 16:32, 12 December 2008 (UTC)[reply]

In [21] I have clarified that 2¹⁰³⁹ − 1 is the record for the special number field sieve algorithm. I added the largest composite Mersenne number factored with proven prime factors and with probable prime factors. Other (unmentioned) algorithms were used but they can only completely factor a number if it happens to only have one large prime factor (the second largest factor can be up to around 30-60 digits depending on some things like the effort put into the factorization). PrimeHunter (talk) 23:09, 12 December 2008 (UTC)[reply]

By the way, 2¹⁷⁴⁷ − 1 has 526 digits. GIMPS [22] only reports the found factors 16811052664235873 and 22831737099099330859999. The cofactor (2¹⁷⁴⁷−1)/(16811052664235873×22831737099099330859999) is composite and has 488 digits. PrimeHunter (talk) 23:29, 12 December 2008 (UTC)[reply]

I see, that makes sense. Thanks for the clarification. — Emil J. 14:49, 15 December 2008 (UTC)[reply]

I think 2ⁿ-1 can always give you a prime IF n = a MERSENNE PRIME I looked at the chart and this seems to work

2²-1=3

2³-1=7

2⁷-1=127

2¹²⁷-1=x

and etc... so if you took the answer to 2¹²⁷-1 and plug that in as the exponent (which means 2^x-1), it should give another mersenne prime. I can't do it but those of you with programs designed for it can probably do so. Hope this helps ^_^ 65.37.24.82 (talk) 06:31, 31 December 2008 (UTC)[reply]

It's not true that ${\displaystyle \scriptstyle 2^{n}-1}$ is prime for all Mersenne primes n: 8191 is a Mersenne prime, while ${\displaystyle \scriptstyle 2^{8191}-1}$ is composite (checked here). But if you would suggest that only your chain: 3, 7, 127, 170141183460469231731687303715884105727, ${\displaystyle \scriptstyle 2^{170141183460469231731687303715884105727}-1}$, etc, consists of primes only, then that might be possible. Checking already the fifth number in that chain would take an awful lot of time though; at around fifty sextillion digits (long scale), it's astronomically larger than the largest known Mersenne prime. If you're able to prove this assertion, you would probably get famous, as you would also have extended the number of known Mersenne primes from finite to infinite (although in reality the new ones established would probably not be very useful for anything). -- Jao (talk) 12:36, 31 December 2008 (UTC)[reply]

Many people have speculated whether the Catalan-Mersenne numbers in your sequence are all prime. Like many other simple to state but hard to solve problems, some amateur mathematicians claim to have proved it but their attempts are rejected by mathematicians. My personal guess is that there are no more primes in the sequence. A few relatively small primes can easily be coincidental (see law of small numbers), and the next numbers are HUGE with sizes where primes are very rare. PrimeHunter (talk) 13:10, 31 December 2008 (UTC)[reply]