Proof that if p is an odd prime, 2p - 1 ≡ 7 or 31 (mod 40).
It can be easily seen that if p is an odd prime, then 2p - 1 ≡ 7 or 31 (mod 40). It can be easily proven that 2p - 1 ≡ 7 or 11 (mod 20). This is because 2p - 1 ≡ 3 (mod 4) and 2p - 1 ≡ 7 or 1 (mod 10). So, in the case that 2p - 1 ≡ 7 (mod 10), 2p - 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 2p - 1 ≡ 1 (mod 10), 2p - 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 2p - 1 ≠ 11 or 27 (mod 40). This is a necessary condition to prove that 2p - 1 ≡ 7 or 31 (mod 40). PhiEaglesfan712 15:52, 13 July 2007 (UTC)
- 24 ≡ 16 (mod 40) and 162 ≡ 16, so 16k = 24k ≡ 16 for all k > 0. Thus 24k+1 − 1 ≡ 16·2 − 1 ≡ 31, and 24k+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, 24k+1 − 1 ≡ 31 (mod 240), and 24k+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, 24k+1 − 1 ≡ 31 (mod 480), and 24k+3 − 1 ≡ 127 (mod 480). This is because for k > 0, 24k+1 − 25 = 25(24k−4 − 1), and 24k+3 − 27 = 27(24k−4 − 1), and the factor (24k−4 − 1) = 16k−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)
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 2p − 1 then 2p ≡ 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 2p ≡ 1 (mod q), raising both sides of the congruence to the power k gives 2kp ≡ 1. Thus 2(q − 1)x ÷ 2kp = 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)
Why is Mersenne not credited for the discovery of any of the primes?
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|
No, he merely conjectured that 2p-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? 184.108.40.206 (talk) 08:42, 3 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...):
- Digitsmp = 1 + int (log102 x p) for Mersenne primes and
- Digitspn = 1 + int (log102 x (2p-1)) for perfect numbers. − Glenn L (talk) 23:33, 13 August 2010 (UTC)
A Recently Disproven Mersenne Prime ?
Anonymous user 220.127.116.11 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 . 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)
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)18.104.22.168 (talk) 04:30, 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").
- 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)
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)
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)
Select[Range[10^3], PrimeQ[2^# - 1] &]
- The sequence you give here are the Mersenne exponents, not the Mersenne primes themselves. Toshio Yamaguchi (talk) 19:27, 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)
- 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 22.214.171.124 (talk) 23:51, 16 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 126.96.36.199 (talk) 15:05, 19 September 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.
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. 188.8.131.52 (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.184.108.40.206 (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.220.127.116.11 (talk) 14:59, 13 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.
18.104.22.168 (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. 22.214.171.124 (talk) 11:03, 25 November 2012 (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)
- 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 (talk) 23:54, 1 February 2013 (UTC) (aka Dubslow)
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.
- I agree. Chmarkine (talk) 20:19, 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)
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 where p is prime. More generally, the numbers of the form are called Mersenne numbers.
- or something along those lines. -- Toshio Yamaguchi 21:41, 4 February 2013 (UTC)
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)
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)
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)
- Google's cache indicates the link  worked 28 November, so let's wait and see if it's only temporarily down. PrimeHunter (talk) 03:51, 4 December 2013 (UTC)
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)