Talk:Mersenne prime/Archive 1

Archive 1

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)

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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

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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)

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Ok, I'll reword it. In fact, "iff" is wrong. If 2^p-1 is prime, then p must itself be prime.

--Axel

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"Iff" being the abbreviation used in logic for "if and only if"?

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Yes.

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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)

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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)

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)

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)

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?

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)

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)

M13 - when found?

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)

revert of a suspicious change

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)

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

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)

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 M_13,466,917 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)

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

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

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)

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)

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)

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

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)

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)

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)

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)

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)

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)

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)

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)

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)

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)

If we denote the recently discovered Mersenne prime by p, then the perfect number associated with it is approximately $(p^{2})/2$, 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.

A bit of original research

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)

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)

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.

Really NOT Mersenne Primes section

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)

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

Codice Palatino

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

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

x^a - y^a divides x^b - y^b implies a|b?

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)

New Mersenne number definition

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)

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

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)

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)

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)

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)

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

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)

Upgrade of the lead

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

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

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)

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)

Word Processor quote

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)

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)

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

My, how embarrassing

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? --Doradus 02:51, 22 September 2007 (UTC)

Missing footnot under "Searching for Mersenne primes"

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)

Missing footnot under "Searching for Mersenne primes"

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)

History

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)

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)

Why there is not an exclusive article for Mersenne numbers?

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)

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)

Distribution of Mersenne primes

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

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

The issue of definition of Mersenne number should get more

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)

"Mersenne gave no indication"?

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)

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)

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)

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

What is the importance of a Mersenne Number?

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)

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)

New record prime - 45th mersenne prime?

(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)

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)

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

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)

Second unverified new Mersenne prime

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)

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

44/46

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)

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)

List updated up to M45,46

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)

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)

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

Generalization?

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)

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)

Writing out number names

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)

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)

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)

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)

Euclid

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)

m31 stated wrongly

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)

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)

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)

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)

Almost always?

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)

factorization "record holders"

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)

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)

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)

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)

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)

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)

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

I think I found a sureproof way

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)

It's not true that $\scriptstyle 2^{n}-1$ is prime for all Mersenne primes n: 8191 is a Mersenne prime, while $\scriptstyle 2^{8191}-1$ is composite (checked here). But if you would suggest that only your chain: 3, 7, 127, 170141183460469231731687303715884105727, $\scriptstyle 2^{170141183460469231731687303715884105727}-1$, 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)

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)

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)

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

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)

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)

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

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)

Why is Mersenne not credited for the discovery of any of the primes?

http://primes.utm.edu/glossary/page.php?sort=MersennesConjecture

n the preface to his Cogitata Physica-Mathematica (1644), the French monk Marin Mersenne stated that the numbers 2ⁿ-1 were prime for

   n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 

Yet the list is as follows:

#	p	Mp	Digits in Mp	Date of discovery	Discoverer
8	31	2147483647	10	1772	Euler
12	127	170141183…884105727	39	1876	Lucas

Should not Mersenne be credited for those two? Alex 68.46.132.117 (talk) 05:25, 3 February 2010 (UTC)

No, he merely conjectured that 2^p-1 was prime for those values, getting two of his four unproved guesses (for p = 67 and 257) incorrect and missing three more (p = 61, 89 and 109). Euler and Lucas actually proved that the above two (p = 31 and 127) were prime. The first seven on his list had already been discovered so don't really count. --Glenn L (talk) 07:01, 3 February 2010 (UTC)

Are you sure the first Mersenne primes were discovered by Greeks, and not Mesopotanians, Egyptians, indians or Chinese? and were is the proof (or reference) of this? 192.87.123.159 (talk) 08:42, 3 May 2010 (UTC)

A Recently Disproven Mersenne Prime ?

Anonymous user 74.3.4.112 noted the following Google Group message:

An amateur mathematician discovered in early 2010 that Mersenne N86,243 is (divisible by 1,627,710,365,249) and N1,398,269.

Nelson & Slowinski originally discovered thenumber and announced it was prime on September 25, 1982. Mersenne Primes Proven Composites?

However, when I tested M86243 on Prime95, I got: "M86243 is Prime! Wd1: 82145A39,00000000"

Although 1,627,710,365,249 = 86,243 * 18,873,536 + 1 and therefore could qualify as a factor, I am very suspicious.

-- Glenn L 11:56, 16 May 2010 (UTC)

The claim was false and has been reverted in [23]. 1,627,710,365,249 is a known factor of 2⁸⁶²⁴³+1. The Mersenne prime is 2⁸⁶²⁴³−1. PrimeHunter (talk) 00:32, 17 May 2010 (UTC)

(obviously) or 30% of the exponent

The number of digits of a mersenne prime is approximately equal to its log base 10, since the number of digits of any number, prime or not, is the one more than the integral part of its log base 10. For example, log(base 10) of 10 is 1, while log(base 10) of 100 is 2.

That the number of digits in a mersenne prime is approximately 30% of its exponent follows from the fact that 2**10 = 1024 ≈ 1000 = 10**3. 10% of the exponent gives the number of 1000's, and three times that gives the number of digits. :-) ( Martin | talk • contribs 05:10, 27 April 2010 (UTC))

You're on the right track. More correctly (remember that log₁₀2 ≈ 0.30102999566...):

Digits_mp = 1 + int (log₁₀2 x p) for Mersenne primes and

Digits_pn = 1 + int (log₁₀2 x (2p-1)) for perfect numbers. − Glenn L (talk) 23:33, 13 August 2010 (UTC)

Mersenne prime number 127 in conversion between inches and centimeters

Please comment on the following valid and important point which has been deleted from the text:

The international inch is defined to be equal to exactly 2.54 centimeters, or equivalently 1 in = 127/50 cm. Thus the Mersenne prime M₇=127 enters conversion between the United States customary units and the International System of Units (SI, often referred to as "metric").

Arcshinus (talk) 02:32, 7 October 2010 (UTC)

As you say, the conversion factor is 2.54. If written as a fraction with coprime numerator and denominator then the numerator happens to be 127. Why is that important to Mersenne primes? It is unrelated to 127 being a Mersenne prime and it isn't even the actual conversion factor. If it should be mentioned anywhere (I don't think it should) then 127 (number) would make more sense. PrimeHunter (talk) 02:57, 7 October 2010 (UTC)

The British-American system of units is still widely used because of historical traditions and industrial machining tools. The system's units such as hand, foot, yard, and fathom are derived by multiplying inch by prime factors 2 and 3 while pace, rod, furlong, and mile introduce prime factors 5 and 11. On the other hand the units in the decimal International System are derived by multiplying by powers of 10 (prime factors 2 and 5). It is remarkable that the conversion between the two system was "rounded" in such a way that a new prime factor 127 appeared. The round-off error distribution statistics is greatly affected by what factors are used in conversion between the systems. So the issue here is more subtle than just being some number. —Preceding unsigned comment added by Arcshinus (talk • contribs) 02:50, 9 October 2010 (UTC)

Wrong Graph?

In the image of the graph showing the digits in the largest known Mersenne prime, why is this graph a line? Shouldn't it be only points at the corresponding points in time when a Mersenne prime was discovered? This way it looks as if new Mersenne primes are continously being discovered, which obviously isn't the case. Toshio Yamaguchi (talk) 14:17, 4 December 2010 (UTC)

The current image is File:Primes.png. I would prefer a step function with vertical connection lines like File:Largest known prime number by year.svg. It seems unclear whether File:Primes.png actually shows the largest Mersenne prime or the largest of all known primes at http://primes.utm.edu/notes/by_year.html. The start looks like 79 digits for a non-Mersenne in 1951. The only other non-Mersenne record is in 1989 with a tiny increase since 1985. But the graph grows smoothly from 1985 (Mersenne) to 1992 (Mersenne), indicating that the 1989 non-Mersenne is correctly ignored. I have posted at User talk:Arvindn#Largest known prime graph. PrimeHunter (talk) 16:56, 4 December 2010 (UTC)

In 1951 the Mersenne record was 77 digits, discovered in 1876. That's the initial point on the graph. If you feel it should be a step function, feel free to replace the graph. It is out of date anyway. Arvindn (talk) 19:16, 4 December 2010 (UTC)

Section on Generalizations

The section "Generalization" seems like it wants to mention the article on repunit primes, but it doesn't do it. It seems like this deserves a note in another section (perhaps "About Mersenne primes"?), but doesn't warrant its own section. Andypar (talk) 05:05, 27 January 2011 (UTC)

I don't understand your point. The section does link to repunit.—Emil J. 11:49, 27 January 2011 (UTC)

For discussion

Mersenne 48 and 49 at OEIS. I have done (in Mathematica) LLT and this:

Select[Range[10^3], PrimeQ[2^# - 1] &]

MartinSojournerfix (talk) 19:25, 10 March 2011 (UTC)

The sequence you give here are the Mersenne exponents, not the Mersenne primes themselves. Toshio Yamaguchi (talk) 19:27, 10 March 2011 (UTC)

Where exactly do M48 and M49 come from? Since when does OEIS list them? Toshio Yamaguchi (talk) 20:41, 10 March 2011 (UTC)

Both the alleged primes are composite with known small factors. Here is PARI/GP proof of the factors 349958939111 and 100313477119 in a small fraction of a second:

? Mod(2,349958939111)^43581437-1
%1 = Mod(0, 349958939111)
? Mod(2,100313477119)^49318327-1
%2 = Mod(0, 100313477119)

Veryfying a real prime of that size would take weeks or months. OEIS doesn't do that. PrimeHunter (talk) 21:21, 10 March 2011 (UTC)

OEIS has now removed the false primes. PrimeHunter (talk) 21:26, 10 March 2011 (UTC)

By the way, the first number has a second known factor:

? Mod(2,5789358091081)^43581437-1
%3 = Mod(0, 5789358091081)

Martin Musatov has a history of this: http://tech.dir.groups.yahoo.com/group/primeform/message/9375. He was banned from submitting alleged primes to the Prime Pages years ago. User:Sojournerfix has been blocked as a sock puppet of User:Martin.musatov. PrimeHunter (talk) 22:27, 10 March 2011 (UTC)

Sophie Germain primes error

Article states: It is also not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from widely believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes.

If p is a SG prime and  ≡ 3 (mod 4), 2p + 1 will indeed divide 2^p - 1, but if p  ≡ 1  this is not true! 89 is SG and 2^89 - 1 is prime. Hence, the claim in the paragraph cited is in error; you need the stronger fact that there are infinitely many SG primes  ≡ 3 (mod 4). —Preceding unsigned comment added by 213.67.74.59 (talk) 23:51, 16 April 2011 (UTC)

Thanks! I have fixed it: [24]. PrimeHunter (talk) 00:06, 17 April 2011 (UTC)

Theorems about Mersenne numbers

I am unsure if this section is appropriate per MOS:MATH#Proofs and I added a cleanup template. The section gives no information about the importance or any other contextual information. I welcome comments from other editors. Toshio Yamaguchi (talk) 17:11, 5 May 2011 (UTC)

Source added for 'Perfect numbers'

I added the citation for Euclid's theorem about Mersenne primes and perfect numbers. He phrases it thus: "If as many numbers as we please beginning from an unit be set out continuously in double proportion, until the sum of all becomes prime, and if the sum multiplied into the last make some number, the product will be perfect." The opening "if" is equivalent to "if we add 1+2+4+8+... to any number of terms", which is another way of saying (...1111) in binary. No idea how to track down Euler's contribution. Grommel (talk) 03:24, 5 June 2011 (UTC)

Ancient Greek mathematicians

I was surprised that my change of "Ancient Greek mathematicians" to "Pythagoras and/or other ancient Greek mathematicians" respective to "Euclid and/or other ancient Greek mathematicians" was reversed and even called "vandalism". The first ancient Greek mathematicians refered to in this case is Pythagoras who was the first known author speaking about "prime numbers" and mentioning 3 and 7 as prime numbers. The second "ancient Greek mathematician" is Euclid who in his book Elementa is the first to mention perfect numbers, which is also mentioned in the wikipedia article on Euclid. Euclid also mention 31 and 127 in the context of perfect numbers. Of course you can argue that it is uncertain what was written by Pythagoras and Euclid. Writings which have their name could have been written by others. It is even uncertain if there ever existed any living persons like "Pythagoras" and "Euclid", or at least someone can question it. If you go to your book shelf though and look up the specific writing where 3 and 7 are mentioned as prime numbers for the first time, this book has the name "Pythagoras" on the cover and if you go for the book where perfect numbers are mentioned for the first time this book is called "Elementa" and has the name "Euclid" on the cover. So I think it is a legitimate opinion to think that their names should be added to the article on Mersenne primes and I disagree with calling it "vandalism". — Preceding unsigned comment added by 193.11.50.158 (talk) 15:05, 19 September 2012 (UTC)

Sources to the findings of M#45 and M#47

When quoting sources a source closer to the actual historical happening is considered better. Nowadays it is often possible to see the traces, relicts, from an occasion on the internet since it is still there.

GIMPS (Global Internet Mersenne Prime Search) is a distributed search for Mersenne primes using softwares like prime95. The computer programs are running on the participants computers and whenever there is a result/output it is sent to a server (primenet-server), where the result is included in the log-file and database. In the GIMPS pressrelease concering Mersenne primes number 45 and 47 it is not mentioned which dates they were found. Fortunately on the user forum, mersenneforum.org, the logfiles (with the faked LL-residues) are quoted: logfile M#45 found on "06-Sep-08 19:53" UTC and logfile M#47 found on "23-Aug-08 7:33" UTC.

All other sources to when these Mersenne primes were found are directely of indirectely based on the information in the logfiles, hence a primary source or relict. 83.216.98.37 (talk) 17:35, 9 October 2012 (UTC)

Please make sure you are familiar with WP:RS. First, Internet fora and other self-published sources are not accepted as reliable sources. Second, Wikipedia articles are supposed to be mainly based on secondary sources, rather than primary.—Emil J. 12:23, 11 October 2012 (UTC)

Anybody can post anything to mersenneforum and most other forums. The forum posts do not make it verifiable that the log files really said that, or that the dates in the log files indicate the original discovery date of the primes. GIMPS shows the discovery date of all their primes at http://v5www.mersenne.org/report_milestones/. That would be a much better source. PrimeHunter (talk) 23:11, 11 October 2012 (UTC)

Since the community participating in mersenneforum.org is the most knowledgeable community on this subject and all of them had access to the logfile at the time we would have had an immidiate reaction to any error or forgery, especially in a situation when something as thrilling as a possible new Mersenne prime was underway. If you read the postings you can see that a large number of persons were actually trying to locate this new result. So any error would not have been unnoticed. And then we would have to ask ourselves: Why on earth would Woltman forge any data in his database but publish the true data on his website? So I cannot find any argument against believing that the quotations of the logfiles are exactly as the logfiles were. Since what Woltman wrote on the webpage is based on the content of the logfiles its more original and primary to refer to the logfiles then to what Woltman wrote based on them, hence a better source.83.216.98.37 (talk) 14:22, 13 October 2012 (UTC)

I would prefer though to add a source showing which date M#45 and M#47 were found. As it is now its unclear where that information comes from.83.216.98.37 (talk) 14:59, 13 October 2012 (UTC)

Which dates were the Mersenne primes #30 and #31 found?

Quite some time ago the date for the find of Mersenne prime #30 was changed from "September 20 1983" to "1983 September 19" and for #31 from "September 6 1985" to "1985 September 1" without giving any reliable sources for these changes. So far I have not been able to find any conclusive arguments for which of the dates are correct. The oldest and, as it appears, most reliable sources have "September 20 1983" and "September 6 1985" respectively, but I don't like to make any changes until I feel I can prove which is right especially since I don't know on which ground the changes were made.

Well, this is just to let you know that I am working on this. Any help is appreciated. 193.11.50.158 (talk) 10:01, 21 September 2012 (UTC)

Found new sorces, solved. 83.216.98.37 (talk) 14:19, 27 October 2012 (UTC)

Which is the oldest source or reference to the first two Mersenne primes 3 and 7 as prime numbers?

Speusippus, c. 408 – 339/8 BCE, wrote a book named On Pythagorean Numbers. This book was mainly based on the work of Philolaus, c. 470–c. 385 BCE, according to Iamblichus, c. 245–c. 325 CE, who obviously had access to both the book of Speusippus and the work of Philolaus and could compare their works. Iamblichus gives us a long, direct quotation of Speusippus and in this quotation we find the oldest known reference to the concepts of prime numbers and composite numbers. It is clear of course that since Philolaus knew about (or "discovered") prime numbers he also knew about the smallest ones like 2, 3, 5, 7, 11.

So why do I also like to include the following passage of the quotation from Speusippus: "Ten does have an equal amount /.../ it is the first in which an equal amount of incomposite [i.e. 1,2,3,5,7] and composite [i.e. 4,6,8,9,10] numbers are seen. /.../ seven is a multiple of none"?

If we take the easiest part first: "seven is a multiple of none", that is a different way of saying that "7 is a prime number". It would be nice to include it since it is the first time ever in the history of numbers that 7 is said to be a prime number. Yes, I only like to quote that small part since its a part of a larger discussion which would only obscure things if we quote.

OK, number 3 then? Is there any reference to number 3 as a prime number. Well, once again if you know that there are prime numbers surely you know that 3 is a prime number. Beside that, in the passage I like to quote, we find a discussion about why 10 is to be recognized as a "perfect number" and here we are not talking about "perfect number" in a modern sense, but the old Greek mathematicians were thinking about numbers with good quality, ideal numbers. So, Iamblichus and my interpretation of Philolaus (according to Speusippus) is that, one of the arguments why 10 should be called a "perfect number" is that among the 10 numbers less than and equal to 10 (1, 2, 3, 4, 5, 6, 7, 8, 9 and 10) we find an equal amount of prime numbers and composite numbers "it is the first in which an equal amount of incomposite and composite numbers are seen." The prime numbers (incomposite) referred to here must be 1, 2, 3, 5 and 7. The composite numbers must be 4, 6, 8, 9 and 10. So, the conclusion from this passage is, even if it is an implicit reference, that Philolaus knew that 3 and 7 are prime numbers.

So the reason why I also want to include these two parts of the quotation:
A. "Ten does have an equal amount /.../ it is the first in which an equal amount of incomposite [i.e. 1,2,3,5,7] and composite [i.e. 4,6,8,9,10] numbers are seen."
B. "seven is a multiple of none"
is that they give a direct reference to the numbers 3 and 7 as prime numbers and its the first time ever they are said to be prime numbers.
83.216.101.203 (talk) 09:57, 25 November 2012 (UTC)

Still meaningless (may be the fault of the translator) and irrelevant. If you want to report the source that 3 and 7 are prime, do so, but not with inappropriate quotes. — Arthur Rubin (talk) 10:09, 25 November 2012 (UTC)

What is it that you do not understand? Please explain because to me the quotations are clear and the interpretations are clear. Yes and on some points the greek text is clearer, lets take the passage: "prime and incomposite numbers, and secondary and composite numbers". If you read the greek original you clearly see that it means "prime and incomposite numbers on one hand, and secondary and composite numbers on the other hand". What makes it difficult is that the ancient greeks thought about numbers in a different way then we do, so you really have to get into their way of thinking before you can understand what they wrote. 83.216.101.203 (talk) 11:03, 25 November 2012 (UTC)

You can note that the source found that 3 and 7 are prime (although not that they are Mersenne primes, because that concept didn't exist), without adding quotes which make no sense in English. — Arthur Rubin (talk) 19:29, 25 November 2012 (UTC)

Suggestion to lock article until Tue, Feb 5th 2013

The few edits that as of now trickled are only a tiny start. Expect a large flood. I suggest to lock article until Tue, Feb 5th (which is known to be the date of the official press release), to save your reverting efforts.

Additional page to consider locking is the "Largest_known_prime_number". — Preceding unsigned comment added by 99.121.250.148 (talk) 20:10, 1 February 2013 (UTC)

I agree. Chmarkine (talk) 20:19, 1 February 2013 (UTC)

You might also want to lock Great Internet Mersenne Prime Search —GraemeMcRae^talk 20:28, 1 February 2013 (UTC)

Articles normally are not being protected preemptively. Articles are normally only placed under protection, if an article previously had been subject to significant vandalism or disruption, (see WP:NO-PREEMPT). Semi-protection, which prevents edits from unregistered users and users with an account that has not yet been autoconfirmed is also only being applied if the article is already the subject of vandalism or disruption. -- Toshio Yamaguchi 20:41, 1 February 2013 (UTC)

I don't think the number of edits regarding the prime are high enough to warrant protection yet. That might change if it intensifies, but we have to wait whether that happens or not. -- Toshio Yamaguchi 20:48, 1 February 2013 (UTC)

Yes, let's wait and see what happens. PrimeHunter (talk) 21:15, 1 February 2013 (UTC)

M48

Regarding M48 which has recently been added, it is being discussed here. Here it is claimed primality has been verified. -- Toshio Yamaguchi 11:09, 27 January 2013 (UTC)

It has not yet been added to the milestones list though. -- Toshio Yamaguchi 11:23, 27 January 2013 (UTC)

Well, I think it's too early to add M48 to this list. It has not been officially verified. Here Prime95 said "Of course, y'all still have to wait for the official verifications." So according to WP:NOTCRYSTAL, I removed M48 from the list. Chmarkine (talk) 01:39, 28 January 2013 (UTC)

Agreed. Lets wait for the official announcement. -- Toshio Yamaguchi 05:23, 29 January 2013 (UTC)

Also note that the first link you posted was not in fact a verification. This, this, and this do count as double checks for official verification purposes. However, the press release will not come until Tuesday morning (when presumably the milestone page will also be updated). Spartan S58 (aka Dubslow) (talk) 23:54, 1 February 2013 (UTC)

Wrong order of definition in lead?

The definition of the article's subject does not appear until the second paragraph of the lead. Wouldn't it be less confusing to start with "A Mersenne prime is a prime number of the form 2^n - 1" and go from there, pointing out that (a) n is necessarily prime and (b) (by back formation?) a number of the form 2^n - 1 is called a Mersenne number? --Vaughan Pratt (talk) 17:59, 4 February 2013 (UTC)

I agree that the lead could be improved. The article starting discussing Mersenne numbers while the title is Mersenne prime is indeed confusing. While it is easy to see the connection by simply reading a bit further, rewriting might help reduce confusion. I suggest to start with

In mathematics, a Mersenne prime is a prime number of the form $M_{p}=2^{p}-1\,$ where p is prime. More generally, the numbers of the form $M_{n}=2^{n}-1\,$ are called Mersenne numbers.

or something along those lines. -- Toshio Yamaguchi 21:41, 4 February 2013 (UTC)

I think this is a good revision. Chmarkine (talk) 03:49, 5 February 2013 (UTC)

I agree it's best to lead with this. We will probably get increased views when a new Mersenne prime is announced later today, so I have changed the lead.[25] PrimeHunter (talk) 04:38, 5 February 2013 (UTC)

It isn't obvious that the exponent needs to be prime though, and that is not part of the definition of Mersenne prime. Rather, that is a theorem about them. So I'm going to remove that line. — Preceding unsigned comment added by 70.199.210.149 (talk) 01:08, 6 February 2013 (UTC)

Infobox Confusion

The Infobox integer sequence at the top of the article is confusing: it looks like Ulrich Regius published on Mersenne primes before Mersenne was born. While this is true, there should be something in the History section to clarify. I'm not knowledgeable enough, but this should be easy for someone who is. By the way, is there an English translation of Regius' work? A Wikipedia article on Regius? I couldn't find either. Myron (talk) 13:56, 6 February 2013 (UTC)

Hmm, I am the one who added the infobox to the article and I also added the statement about Regius there. I think the reason I added this is because of the table on page 39 (if you download it as PDF file, it is the 39th page of the PDF file). The second row of the table exactly lists the first five Mersenne primes. I can't read Latin, so I am not sure what exactly the purpose of that table is. The first row lists the powers of two and the third row lists the perfect numbers. So it seems to me he studied perfect numbers (which, as our article says, were already known to Euclid in the 4th century BC) and in that process also discovered the Mersenne primes, thus beating Mersenne by more than 100 years (see also http://primes.utm.edu/mersenne/index.html). primes.utm.edu says Regius was the first to show that (contrary to a belief at that time) the numbers of the form 2ⁿ-1 (the Mersenne numbers) are not all prime. I am not aware of an English translation of Regius work. (At least I couldn't find one via a quick Google search, but that doesn't necessarily mean that none exists). -- Toshio Yamaguchi 16:08, 6 February 2013 (UTC)

The pronounced tendency of the smallest Mersenne numbers to be prime was recognized in antiquity. Regius may have been first to publish at all systematically on the topic and to have demonstrated in print that not every Mersenne number is prime, supposedly a surprise to scholars, although it is hard to believe that at least several people had not earlier managed to factor M₁₁ into 2047 = 23*89, something that can be done by trial division in not much time. An earlier (1456, author anonymous?) manuscript is said to have shown that M₁₃ is prime (see http://page.math.tu-berlin.de/~kant/Mersenne/mersvortrag.pdf). Where is that manuscript, did it rely on trial division, how did that help develop the field? Was it because this was at the time the greatest Mersenne number actually found to be prime? Mersenne, drew attention by finding factors for several 2ⁿ - 1 numbers and by generally extending the assertion up to n = 257, leading subsequent authors to attach his name. The History section should detail this and post citations. Also, Regius' Utrisque arithmetices epitome appears to be a major work, considering that reprints are currently available in paperback (!), but I can't read Latin either (or even write it). Regius deserves a Wikipedia page showing why his book matters, or mattered, if it really does, or did. Myron (talk) 18:59, 6 February 2013 (UTC)

I don't see how Regius belongs in the infobox. He wasn't first to consider Mersenne primes (Euclid proved they generate perfect numbers) and he didn't name them. Most readers will assume at least one of two happened in 1536 by Regius when the infobox says "Publication year 1536. Author of publication Regius, H." It's possible (I don't know) that Regius did the most significant work on Mersenne primes at the time, but he is still just one in a series of people who have studied them. PrimeHunter (talk) 03:12, 7 February 2013 (UTC)

Then who should be listed in the infobox? I mean, there must be a first publication investigating Mersenne primes, and that's why I included those two parameters in the infobox: that one can quickly see when the first publication about the specific numbers appeared and who is the author of it. Maybe we should list Euclids work instead then? According to Euclid's Elements#Contents of the books he discusses them in his book no 9, although Euclid's Elements#Basis in earlier work says most of the books are theorems proved by other mathematicians, so there might be an even earlier work discussing them. -- Toshio Yamaguchi 09:52, 7 February 2013 (UTC)

Of course we could also simply just remove that info from the infobox entirely or maybe change something in the infobox, if another parameter would be more fitting to hold that information (if it should appear in the infobox at all). I will wait for feedback of what others think, before doing anything. -- Toshio Yamaguchi 13:01, 7 February 2013 (UTC)

But ...

This may be a dangerous question to pose, but it would be nice to know (ie. add to article) what practical use is or can be made of Mersenne numbers, if any. I get that the search is 'fun' (fsvo) in itself but are there specific use cases for this series of numbers? --AlisonW (talk) 22:48, 7 February 2013 (UTC)

Some applications are listed at the bottom of the section Mersenne prime#About Mersenne primes. -- Toshio Yamaguchi 09:55, 8 February 2013 (UTC)

Unifying representations of numbers

I noted there is some inconsistency in the representation of the numbers in this article. For example long numbers in Mersenne prime#History and Mersenne prime#Factorization of composite Mersenne numbers use comma separated digit groups, while the numbers in Mersenne prime#List of known Mersenne primes don't use commas. Wikipedia:Manual of Style/Dates and numbers#Delimiting (grouping of digits) says numbers with five or more digits should be separated into groups using commas and also says that in scientific articles thin spaces can be used instead. Which style should be used in this article? I suggest to apply that style to all numbers in this article, after an appropriate one has been identified. -- Toshio Yamaguchi 15:14, 9 February 2013 (UTC)

Generalization section

The text there matches from this reference: 1. It may or may not be appropriate here but would need a citation at the least. (I researched it because of the edit mark at the start of the section made May 2011].--Billymac00 (talk) 00:53, 11 February 2013 (UTC)

That's a mirror of Wikipedia. They even pull the images from our servers but don't give us the credit required by our license and Wikipedia:Reusing Wikipedia content. PrimeHunter (talk) 05:13, 11 February 2013 (UTC)

Proof that if p is an odd prime, 2^p - 1 ≡ 7 or 31 (mod 40).

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)

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)

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)

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)

The talk page is for suggesting improvements to the article, not for discussing the subject of the article. Blackbombchu (talk) 03:33, 4 December 2013 (UTC)

Dead link

The 3^rd reference is a dead link. Blackbombchu (talk) 03:29, 4 December 2013 (UTC)

Google's cache indicates the link [26] worked 28 November, so let's wait and see if it's only temporarily down. PrimeHunter (talk) 03:51, 4 December 2013 (UTC)

The link works now. PrimeHunter (talk) 21:28, 7 December 2013 (UTC)

Consistency

Sometimes Wikipedia has 2 different ways to write an article an despite that neither of them is enforced throughout the entire Wikipedia system, each individual article is supposed to pick only one of the 2 styles to stick to and not mix them, for example Wikipedia's policy doesn't allow a ship to be refered to as it in one part of an article and she in another part of the same article. For consistencey, since most of the mathematical expressions that are not not their own sepearte line are using html code, I think the rest of the mathematical expressions in the article that are not on their own separate line should also be switched from latex to html code. Furthermore, I know the html code for those expressions really well so I should be the one to make that change. Is it fine for me to make that change, only for the ones that are not by themselves on a line? Blackbombchu (talk) 20:40, 7 December 2013 (UTC)

@Blackbombchu: As far as I know there is no rule to when to use HTML and when to use TeX. Many formulas in this article already use HTML, so I think those that appear as intended in HTML should be changed from TeX to HTML. The only case where I think using TeX has a clear advantage over using HTML is in formulas containing vertical lines (such as fractions or the root symbol), in cases where the HTML symbols look really ugly (such as $\mathbb {Q}$) or where the correct meaning of a formula becomes much harder to grasp when written in HTML or is not expressible at all (such as in complicated integrals or other formulas where symbols are stacked vertically). -- Toshio Yamaguchi 13:25, 11 April 2014 (UTC)

2^P-1 ≡ 1(mod 6p)

2^P-1 is an odd number ⇒ 2^P-1 ≡ 1(mod 2)

By Fermat's little theorem, we see that, 2^P ≡ 2(mod p) ⇒ 2^P-1 ≡ 1(mod p)

if p is an odd prime then: 2 ≡ -1(mod 3) ⇒ 2^P ≡ (-1)^P(mod 3) ⇒ 2^P-1 ≡ -2(mod 3) ⇒ 2^P-1 ≡ 1(mod 3)

So we got:

2^P-1 ≡ 1(mod 2)

2^P-1 ≡ 1(mod 3) ... ( for p>2 )

2^P-1 ≡ 1(mod p)

So, for p>3 , we can found that 2^P-1 ≡ 1(mod 6p)

Note: 2^P-1 doesnt have to be a prime number!

Isaac.mor (talk) 08:30, 24 October 2014 (UTC)

The last 3 digits of 2ⁿ-1

Mn = 2ⁿ-1

its well known that you can build Mn digits using only the digits of n

lets show a few examples:

if the last 2 digits of n are ....17 then Mn last 3 digits have to be ....071

if the last 2 digits of n are ....23 then Mn last 3 digits have to be ....607

i will only show the roles for an odd n because we wanna use it for primes

p Mp ... 03 ... 007 ... 23 ... 607 ... 43 ... 207 ... 63 ... 807 ... 83 ... 407 ... 07 ... 127 ... 27 ... 727 ... 47 ... 327 ... 67 ... 927 ... 87 ... 527 ... 11 ... 047 ... 31 ... 647 ... 51 ... 247 ... 71 ... 847 ... 91 ... 447 ... 19 ... 287 ... 39 ... 887 ... 59 ... 487 ... 79 ... 087 ... 99 ... 687

or

p Mp ... 09 ... 511 ... 29 ... 911 ... 49 ... 311 ... 69 ... 711 ... 89 ... 111 ... 01 ... 751 ... 21 ... 151 ... 41 ... 551 ... 61 ... 951 ... 81 ... 351 ... 17 ... 071 ... 37 ... 471 ... 57 ... 871 ... 77 ... 271 ... 97 ... 671 ... 13 ... 191 ... 33 ... 591 ... 53 ... 991 ... 73 ... 391 ... 93 ... 791

just so you know the same works for 3 digits of n lets show a few examples:

2^{... 639}-1 = 2^{... 139}-1 = ... 1887

2^{... 711}-1 = 2^{... 211}-1 = ... 8047

etc ...

the same works for any k digits of n

but for really big numbers you need a LOT of computer power :)

Isaac.mor (talk) 12:03, 24 October 2014 (UTC)

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New Largest (Mersenne) Prime Found

A new Mersenne prime was discovered on Jan. 07 2016. See http://www.mersenne.org/primes/. This calls for a substantial edit to this page as well as many other wikipedia pages, because "largest prime", "second largest prime", and "largest mersenne prime" are used extensively referring to now-incorrect numbers.

66.31.237.80 (talk) 05:34, 24 January 2016 (UTC)

The new record has already been added to more than a dozen articles including several mentions in this one. If you know of a specific article or place that still needs an update then please post it. PrimeHunter (talk) 06:06, 24 January 2016 (UTC)

Graph issue

I think the graph in the article is wrong. There is a slow increasing line between 1950 -1960 indicating every year a bigger Mersenne prime was found with an increase in size equal to the previous each year, then between approx. 1960 -1962 for two years the rate increased. in my opionion there should be horizontal lines between discoveries of the finds and no steady increase. for the last few years it does not matter that much as the rate of finds are increased so much that the graph will be about the same (i think the point of the graph)195.240.149.123 (talk) 04:30, 13 August 2010 (UTC)

I think one of the words is wrong:

I think that the plot is a "logarithmic scale" and not a "double logarithmic scale". The term semi-log plot

The x-axis of time is linear for the year, but the y-axis for number of digits is the base 10 log. Peter10003 (talk) 18:41, 29 July 2016 (UTC)

You are correct. The scale on the y-axis is a logarithmic scale not a double logarithmic scale. If it were a double logarithmic scale the values corresponding to those actually shown would be 2, 3, 4, 5, 6 and 7. The type of graph is a semi-log graph - specifically a log-lin graph. I have changed the wording on the graph. AirdishStraus (talk) 12:22, 30 July 2016 (UTC)

No, if you actually read the caption you will see that it (1) does not say that the graph is double-log scale, and (2) is completely correct. --JBL (talk) 13:51, 30 July 2016 (UTC)

Apologies if I didn't make myself clearer in the previous edit. The y-axis is a logarithmic scale of the number of digits of the prime number but is a double logarithmic scale (rounded down to the nearest power of 10) of the value of the prime number. AirdishStraus (talk) 14:08, 30 July 2016 (UTC)

I don't know where you got the idea that there is rounding being done; as far as I can tell this is not true. --JBL (talk) 14:12, 30 July 2016 (UTC)

The number 2?

I freely admit to not being a mathemetician, or indeed particularly number-savvy with regard to primes, but there is an inconsistency in the article with regard to the number 2. Within the confines of this article, is "2" considered a Mersenne prime? Part of the article suggests it is, and part not.

In the history section, 2 is listed as a prime (and the section states that "His list was accurate through 31",) and the image includes 2 as a Mersenne prime, yet the rest of the article - especially the lede: "The first four Mersenne primes (sequence A000668 in the OEIS) are 3, 7, 31, and 127." - doesn't include 2.

I see that there is reference to two different OEIS sequences - one of which includes 2, and one that doesn't, however this is confusing to those who don't have an in-depth understanding of the subject matter. In short - as it currently stands the article is inconsistent, and should:

1. the number "2" be included in the lede as the first five Mersenne primes,
2. the rest of the article be adjusted to remove the number 2?
3. a small explainer be added in somewhere to clarify the inconsistency, and why it is and isn't included in different parts of the article? Chaheel Riens

(talk) 06:28, 1 January 2017 (UTC)

The sequence containing the "2" refers to the exponent (power) in M_p = 2^p − 1. None of the sequences say that "2" is a 'Mersenne prime'. AirdishStraus (talk) 10:35, 1 January 2017 (UTC)

Thanks for replying, but you're kind of proving my point. I look at that and go "What?" If it's not a Mersenne prime, why is it listed in the History section, and why is it highlighted in the image? (The image, incidentally does say that it's Mersenne prime: "The first 64 prime exponents with those corresponding to Mersenne primes shaded in cyan and in bold".) I'm not picking a fight here, I'm just pointing out an unexplained discrepency in the article. Chaheel Riens (talk) 11:21, 1 January 2017 (UTC)

2 is a Mersenne prime exponent but not a Mersenne prime. The exponent is p in M_p = 2^p − 1. The Mersenne prime with exponent 2 is M₂ = 2² − 1 = 3. It's cumbersome and unnecessary to write a list of Mersenne primes (or alleged Mersenne primes) as 2² − 1, 2³ − 1, 2⁵ − 1, 2⁷ − 1, 2¹³ − 1, 2¹⁷ − 1, 2¹⁹ − 1, 2³¹ − 1, 2⁶⁷ − 1, 2¹²⁷ − 1, 2²⁵⁷ − 1. The exponents are usually listed instead: 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257. PrimeHunter (talk) 12:02, 1 January 2017 (UTC)

Ok, I get it - just, but the history section states that the number "2" is a "Mersenne Prime", not an "Mersenne Prime exponent". that needs addressing then. Chaheel Riens (talk) 12:25, 1 January 2017 (UTC)

Where does it claim that? I see only "The exponents listed by Mersenne were as follows: 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257." --JBL (talk) 16:24, 1 January 2017 (UTC)

In the table description: "The first 64 prime exponents with those corresponding to Mersenne primes shaded in cyan and in bold". Again, I hold my hand up and say I'm no expert of any stretch in this field, but given that the number 2 is both shaded, and in bold - the associated text - "corresponding to Mersenne primes" - suggests that 2 is a Mersenne prime. This may just be a poor description that needs tweaking. Chaheel Riens (talk) 16:46, 1 January 2017 (UTC)

What that says is that 2 is a prime exponent, and that it corresponds to (not "is") a Mersenne prime. But mathematical English is not always clear to those not fluent in it; do you have a suggestion for rewording? --JBL (talk) 17:02, 1 January 2017 (UTC)

Not yet, but now that it's been explained to me, I'll certainly think about. I solidly fall into the "not fluent" category, and it puzzled the hell out of me. Chaheel Riens (talk) 18:29, 1 January 2017 (UTC)

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Unsourced "pyramid charts" addition by Kulprit001

Does anyone understand what [27] means? It's not a minor edit, it's unsourced, it's been undone by four different editors, and User:PrimeHunter has been unable to find anything about the claim [28]. Meters (talk) 04:28, 3 December 2017 (UTC)

It is not a claim, it is a fact that is useful for generating pyramid charts. You wont find out by googling it, you will find out by doing it. — Preceding unsigned comment added by Kulprit001 (talk • contribs) 04:42, 3 December 2017 (UTC)

My addition of " Can also be useful for calculating the space in between points on a pyramid chart as well as the space surrounding it, where n represents the amount of levels of divulgation." to the /* /* Mersenne numbers in nature and elsewhere */ is continuously challenged. However it is a provable mathematical fact, simply open up any image editor and draw any series of pyramid charts as pixels to properly illustrate the point. If you do not believe me try it for yourself. draw any pyramid chart, and count the spacing between between points and inversely the space surrounding the points and you will see that for n levels of dilvulgation, the spacing can be calculated with the Mersenne formula. If anyone want to change this again, please tell me that you have tried this for yourself. If so and you still do not believe me, than i would suggest that you think about it for a bit longer, before immediately dismissing it again. — Preceding unsigned comment added by Kulprit001 (talk • contribs) 04:30, 3 December 2017 (UTC)

We're asking your for an independent reliable source to verify the claim. See WP:RS. Original research is not acceptable See WP:OR. Continuing to edit war over this is not a good idea. Meters (talk) 04:37, 3 December 2017 (UTC)

Your right, it is not a good idea for you, but it is a good idea for the page. As a great man once said, "if you cannot see what is right in front of you then you are indeed a fool." Please physically investigate this for yourself before any further complaints. this is a useful addition not a reckless destruction, so please do not treat it as such. — Preceding unsigned comment added by Kulprit001 (talk • contribs) 04:47, 3 December 2017 (UTC)

Please stop the vadalism, or i will have to email wikipedia about this. i am making a valid useful contribution anyone who checks it out for themselves will understand on be on my side. i cannot provide a source for nature and basic geometry so i am sorry, but your just going to have to use your eyes and brains. — Preceding unsigned comment added by Kulprit001 (talk • contribs) 04:58, 3 December 2017 (UTC)

Following Wikipedia's established guidelines is not vandalism. Why are you not willing to read the policies other editors have referred you to in their edit summaries, and follow them? Your edit needs a source -- Wikipedia is not meant for original ideas/discoveries, even if they are correct. You are not allowed to revert more than 3 edits within 24 hours, but you did this 7 times already. This behavior could result in a block. Gap9551 (talk) 05:01, 3 December 2017 (UTC)

If i get blocked than that just proves that wikipedia is a useless pile of rubbish and that indeed the vast majority of people are in fact not intelligent at all, but petty belligerent fools. — Preceding unsigned comment added by Kulprit001 (talk • contribs) 05:07, 3 December 2017 (UTC)

After the comments by several editors, has the possibility occurred to you that you may misunderstand the purpose of Wikipedia? It gives an overview of established facts using sources; it is not a place to publish facts that lack such sources. Sources are not only needed to verify the correctness of facts, but also to show that these facts are important enough for inclusion in Wikipedia. Gap9551 (talk) 05:12, 3 December 2017 (UTC)

wikipedia, a pun for world wide web encyclopedia. encyclopedia, a book giving information. My addition is information that is immediately verifiable and self evident, and is therefore viable information. how is Wikipedia ever to improve if it seeks to remove and destroy basic facts? then it would be called wikibook. so i am sorry but you are all wrong. please read my addition more carefully, try it for yourself and ponder it for at least a day before even thinking about removing it as the is no logical or reasonable basis for doing so. — Preceding unsigned comment added by Kulprit001 (talk • contribs) 05:23, 3 December 2017 (UTC)

Actually, your addition is not clearly stated. "pyramid chart" is not defined or linked to. The "points on a pyramid chart" are not identified. Any 2D chart contains an infinite number of points so you'd need to define which you mean. "the space surrounding it" is vague. "Can also be useful" is vague, it does not say how exactly it would be useful. The meaning of "levels of divulgation" is not defined or linked to. But even if all this would be clarified, you'd still need a source. Gap9551 (talk) 05:28, 3 December 2017 (UTC)

PLEASE STOP UNDOING! there is no need to. it does not change anything, it only adds an additional valid & verifiable point, which is in the spirit of wikipedia. It is a description of an image therefore it require no citation. please read and understand the rules before attempting to enforce them. — Preceding unsigned comment added by Kulprit001 (talk • contribs) 05:28, 3 December 2017 (UTC)

PLEASE STOP UNDOING! there is no need to. it does not change anything, it only adds an additional valid & verifiable point, which is in the spirit of wikipedia. It is a description of an image therefore it require no citation. please read and understand the rules before attempting to enforce them. Draw a pyramid chart, count the spacing, if you still do not understand than i don't think you have any authority to undo it, because you obviously don't understand it therefore have no right to comment.— Preceding unsigned comment added by Kulprit001 (talk • contribs) 05:30, 3 December 2017 (UTC)

It's not a description of an image or chart. As has been pointed out, there is no image to describe. And what you are doing is making a claim about how Mersenne primes numbers can be used to analyse a particular type of chart, not describing the chart. Since you say on your talk page [29] that you have written an app that uses Mersenne primes numbers to do this, it appears that this is WP:OR. Meters (talk) 09:08, 3 December 2017 (UTC)

@Kulprit001:, the belligerent and pig-headed way you approached this discussion was always doomed to failure: Wikipedia works on a collaborative model, and if you can't explain to others what you're doing politely then it will never work. That being said, @Everyone else: Kulprit is almost certainly trying to express that the number of nodes in a full (or complete) binary tree with n layers is the Mersenne number 2^n - 1. (Or something equivalent.) This is a totally true thing. And in fact Mersenne numbers are a common answer to lots of enumerative combinatorial questions, although they aren't usually called "Mersenne numbers" in that context. It is a reasonable question about whether these combinatorial facts should be listed somewhere, either in the section Kulprit was trying to add to, or in a separate subsection called "in enumeration" or something. --JBL (talk) 14:37, 3 December 2017 (UTC)

If that is what Kulprit001 is trying to say, then yes it correct. If we want to include it then we need to express it in some manner that is comprehensible though. My opinion is that mentioning Mersenne numbers for complete binary trees alone is not warranted. As Joel B. Lewis points out, this relationship shows up in many enumerations. There's nothing special about binary trees. The solution to the Wheat and chessboard problem is exactly the same progression. for example. If we want to include this trivial relationship just mention it as the result of summing a geometrically doubling series starting at 1. The number of nodes in a complete binary tree can be listed as one of the examples. Meters (talk) 19:54, 3 December 2017 (UTC)

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Mersennes in nature and pernicious

I removed the following text from the introduction:

equivalently that they be pernicious Mersenne numbers, namely those numbers whose binary representation contains a prime number of ones and no zeros

and this from "Mersenne numbers in nature":

In computer science, unsigned n-bit integers can be used to express numbers up to M_n. Signed (n + 1)-bit integers can express values between −(M_n + 1) and M_n, using the two's complement representation.

Both are based on the belief that a Mersenne number is 2ⁿ−1, which is a mistake. The correct "pernicious" part is that there be a prime number of 1's followed by a large number of 0's that seems too trivial to bother mentioning (especially in an introduction). Zaslav (talk) 04:13, 6 October 2018 (UTC)

Incomprehensible paragraphs

The section about "primitive part" is impossible to understand to the largest part of it. For example,

Besides, if we notice those prime factors, and delete "old prime factors", for example, 3 divides the 2nd, 6th, 18th, 54th, 162nd, ... terms of this sequence, we only allow the 2nd term divided by 3, if we do, they are ...

The phrase is grammatically incorrect and incomplete, and it is not clear what the author wanted to say. It's similar for the whole subsection and a later one, probably from the same "contributor". Is anyone please willing to improve this? Such "contributions" are annoying, the article would be better without it. I'd suggest to move the subsection here (i.e., delete it from the main page) until it is rewritten in correct English. As it stands, it barely qualifies for a comment on this talk page. But I don't know whether doing so is (WP-)"politically correct". — MFH:Talk 17:48, 5 December 2018 (UTC)

@MFH: I agree and have removed the section. Which other section(s) have the same problem, in your estimation? --JBL (talk) 18:26, 5 December 2018 (UTC)

Incidentally, the section is at least 3 years old (I got bored of looking back further). --JBL (talk) 18:33, 5 December 2018 (UTC)