Talk:Largest known prime number/Archive 1

Archive 1

Initial text

The original article presented the number, or what purported to be the number, or, come to think of it, just a long string of digits. This added up to close to four megabytes, and the only reason I can think of for presenting it would be that some people would say "Wow". That is, if they hadn't got bored waiting for the download, and if their browsers hadn't crashed. Suggestion: if the string of digits is important, provide a link to it on some other page. -- Hoary 11:36, 2005 Mar 30 (UTC)

Yes, I agree completely. Good move, Hoary. BTW, did anybody check that number for correctness? I think the 3'588'172nd digit was off by one... :-) Lupo 11:42, 30 Mar 2005 (UTC)

I can't find the link to it. Alphabetagamma 02:40, 28 September 2006 (UTC)

I give up! Verification of prime numbers take too long... (pun intended)

I tried checking it but I figured I'dd leave it to the Kool people at Karleton U.(it's actually a "C")(I'm their rival from Ottawa U.) I tried thinking of a way to include this information without to much contreversy. Anyone here, a regular editor, want to venture into adding the information about the verification of the number? The information is at http://www.magazine.carleton.ca/2004_Fall/1342.htm --CyclePat 00:58, 7 January 2006 (UTC)

p.s.: yes! someon has tried to verify it. --CyclePat 00:58, 7 January 2006 (UTC)

and the winner is... ? (But what exact number are they talking about, anyway ? ;-) --FvdP 02:03, 7 January 2006 (UTC)

To verify that :2^32,582,657 − 1 is really a prime number you have to use the Lukas-Lehmer method. It ll take more than 3 months in a P4. -- Magioladitis 17:55, 16 April 2007 (UTC)

Date confusion

According to the article, the largest prime "as of January 2007"..."was confirmed to be a prime number on September 11, 2006." So for the last few months of 2006 there were others that were larger but got decertified? DMacks 04:41, 20 February 2007 (UTC)

"as of January 2007" means the latest date the information was known to be current when the edit was made. It's February now and the record has not been broken, so an edit today could say as of February. See Wikipedia:As of for the use of the notation. PrimeHunter 11:58, 20 February 2007 (UTC)

Ah that makes wiki-technical sense. It's a linguistic mess though. How about "The largest prime [[As of 2007|presently]] known..."? DMacks 16:34, 20 February 2007 (UTC)

You can suggest changes at Wikipedia talk:As of. The idea is for the reader (including non-editors) to see when the information was known to be correct. I think the text should indicate that. The URL is not enough, because many will not see it. An article claim that something is "present" can quickly become wrong when the article is not updated immediately after a new event. There are lots of existing links saying "as of year" or "as of month year" and it seems easier to locate and maintain them with a standardized name, e.g. visible in the backlinks at Wikipedia:As of. I don't want to change the established name without consensus, and I wouldn't support any change which doesn't display the year. Note that some "as of year" links were added with a former year (e.g. from an old source) without knowing the status at the time of the edit. PrimeHunter 20:59, 20 February 2007 (UTC)

My suggestion was simply to change the text displayed in the article so that the sentence makes more sense, not change away from using the template nor change the page to which the link points. DMacks 21:11, 20 February 2007 (UTC)

Proven by Euclid

True, but misleading. I've never heard of any evidence he was the 1st to prove it. Peter jackson (talk) 10:48, 19 September 2008 (UTC)

46th or 45th mersenne prime?

I changed the text from 45th to 46th based on a UPI.com article that I referenced, but I think that the original number, 45th, may be correct. The number (in this case 45 or 46) must be a count of the numbers identified as the largest Mersenne prime known to date. One of the articles notes that a smaller Mersenne prime was found two weeks after the one with 12,978,189 digits. ( Martin | talk • contribs 06:21, 27 May 2009 (UTC)) This page [1] lists the largest known Mersenne primes, but the prime listed at number two was found (I believe) after the largest prime was found, and so would not be assigned a number in the list of Mersenne primes, would it? ( Martin | talk • contribs 06:25, 27 May 2009 (UTC))

Two numbering systems are in use: In order of discovery among all Mersenne primes (not just among those which were the largest known prime at the time), and in order of size among all known Mersenne primes at a given time (this number changes if a smaller Mersenne prime is found later). The largest known prime 2^43,112,609−1 was the 45th in order of discovery and this is what "chronologically" refers to so I will revert your edit [2]. There are currently 46 known Mersenne primes in total. The 46th in order of discovery is the 45th in order of size, and the 45th in order of discovery is currently the 46th and largest in order of size. There are still many untested Mersenne numbers below both of them so more smaller primes might be found. By the way, some cranks have made selfpublished claims to invent new methods to prove primality of Mersenne numbers far above the largest known prime (the Double Mersenne number 2²¹²⁷⁻¹−1 is a popular crank target), but their claims are original research and rejected by professional mathematicians and reliable sources, so Wikipedia correctly ignores them. PrimeHunter (talk) 10:35, 27 May 2009 (UTC)

other GIMPS discoveries

Is it really necessary to mention the discoveries of M45 and M46? (by size, not discovery date, i.e. M37156667 and M42643801) Neither of them was ever the largest known prime number, which is the topic of this article. Their discoveries had nothing to do with the largest known prime number. — Mini-Geek ^(talk) 17:54, 18 September 2009 (UTC)

Including them seems OK to me. There is only one line about each. They are the second and third largest known primes. Some readers may think the latest Mersenne prime found by GIMPS is the largest. The article explains this is not the case. PrimeHunter (talk) 19:37, 18 September 2009 (UTC)

Including history of largest known prime number

What about including a section on the history of the largest known prime number in order to give this article a bit more content? What I have in mind is for example making a table giving the year of discovery, number of digits and maybe the name of the discoverer(s). Ideas and comments are welcome. Toshio Yamaguchi (talk) 11:29, 11 April 2011 (UTC)

Mmmh my above proposal seems to be partially redundant, since actually thats what the current table contains anyway, yeah. But what about expanding that table a bit into the past in order to have a bit more content in this article? Toshio Yamaguchi (talk) 15:23, 11 April 2011 (UTC)

I started a table showing the development of the largest known prime number over the years. Toshio Yamaguchi (talk) 01:11, 17 April 2011 (UTC)

Error

The table of the largest known prime number seems to have been altered recently. For example, it now contains M_42643801, which never was the largest known prime number. While I think the article should mention it as the 2nd largest known prime number, it appears to be misplaced in the table, which only lists the largest known prime numbers by year. M_42643801 was never the largest known prime number, which is the topic of this article. Therefore its inclusion in the table doesn't make sense to me. Toshio Yamaguchi (talk) 17:16, 24 April 2011 (UTC)

Reversion of last three edits

I reverted the last three edits. None of the newly added primes have ever been the largest known prime number. Toshio Yamaguchi (talk) 17:22, 24 April 2011 (UTC)

an idea of finding a larger prime number

ok prime = 2^p − 1 where p is also a prime number...

so there for a larger prime number would be 2^{243,112,609 − 1} − 1

if not, why? James137 (talk) 01:07, 18 May 2011 (UTC)

also heres a larger prime number to the acticle (2^10000000019) - 1 :-D

James137 (talk) 01:13, 18 May 2011 (UTC)

The first number you describe is a Double Mersenne number. Not each Double Mersenne number for a given Mersenne prime exponent (the exponent of the Double Mersenne prime being a Mersenne prime) is necessarily itself a prime number. For example, while the Mersenne number M₃₁ is a proven prime number, the Double Mersenne number M_M31 is composite. Therefore, while 2^43112609 − 1 is proven to be prime, this does not necessarily mean that 2^{243,112,609 − 1} − 1 is also prime. It is a probable prime, but it might be possible, that a factor will be found in the future.

Your second number is also a probable prime. That does not mean it is prime. It might turn out that it is really prime or that it is composite, therefore its status is probably prime, which also means its exact status is unknown and any statement in either direction without a proof is therefore speculative. Toshio Yamaguchi (talk) 12:03, 19 May 2011 (UTC)

@Toshio: In what sense are these two numbers "probable primes"? They're certainly not listed at [3]. I'm not aware of a primality test able to test such high numbers at present. That would be an interesting result, since the vast majority of Mersenne numbers are not primes, after all... --Roentgenium111 (talk) 21:01, 19 May 2011 (UTC)

@James137: The claim in your first sentence is wrong - not every number of the form 2^p-1 with p a prime is itself prime. Read the article Mersenne prime:

"While it is true that only Mersenne numbers M_p, where p = 2, 3, 5, … could be prime, often M_p is not prime even for a prime exponent p. The smallest counterexample is the Mersenne number

M₁₁ = 2¹¹ − 1 = 2047 = 23 × 89,

which is not prime, even though 11 is a prime number." --Roentgenium111 (talk) 21:16, 19 May 2011 (UTC)

@Roentgenium111: I used the term "Probable prime" here to indicate that their complete factorizations are currently not known. Actually this is not consistent with the definition of this term and I realize it is therefore incorrect. The purpose of my statement was to indicate that I am not aware of any single factor of either 2^{243,112,609 − 1} − 1, or 2^10000000019 − 1, but that not knowing any factors of a number that has not been proven prime is a proof of neither is primality, nor its compositeness, thus both possibilities remain open. Toshio Yamaguchi (talk) 21:34, 19 May 2011 (UTC)

I see. But even independent of the exact definition, since these numbers are probably not primes (as I explained above), you should rather call them "improbable primes", or maybe "possible primes". --Roentgenium111 (talk) 13:42, 20 May 2011 (UTC)

I think it is rather irrelevant if we called them "Probable prime" or "Improbable prime", since it is effectively the same. A number that possibly might be prime might also possibly not be a prime, so this is all equivalent to saying its actual property is unknown. I like your suggestion of "Possible prime", but unless this term has actually been used in any mathematical publications, using it on Wikipedia would be kind of original research. Toshio Yamaguchi (talk) 14:06, 20 May 2011 (UTC)

It's far from irrelevant. The term probable prime has a fairly specific meaning: A number which has passed one or more probable prime tests (prp tests). The term is usually used for numbers which have not yet been proved prime, but are statistically very likely to be prime (often more than 99.999%) based on the used prp test(s). The discussed numbers have not passed a prp test and are very unlikely to be prime so it's false to call them probable primes now. PrimeHunter (talk) 19:12, 5 February 2013 (UTC)

Yes, I agree. I am aware now that there are probabilistic and deterministic primality tests and that a number that has passed the first type is a prp, while a number that has passed the second type is actually prime. For example I guess that if a randomly chosen large integer has passed like a few hundred Fermat primality tests in different bases, then it is very likely to be prime and therefore a prp, since Carmichael numbers become very rare when considering large numbers. -- Toshio Yamaguchi 19:54, 5 February 2013 (UTC)

Or more precisely the pseudoprimes in any base get very scarce as numbers get large I guess. There might exist numbers that are Fermat pseudoprimes in a number of small bases but fail to be a pseudprime in some large bases, so actually what seems to be relevant is the scarcity of pseudprimes, of which the Carmichael numbers are a subset. -- Toshio Yamaguchi 20:08, 5 February 2013 (UTC)

M48 edits need a bit more work

The new hybrid paragraph (The first sentence about M48 and the rest about $100,000 prize) is factually incorrect. Only the first 10-million digit number was eligible for the said prize. Either the mention of the old record holder should be retained in connection to the prize, or the prize part should be dropped.

While the page needs to be updated to mention the current record, the mention that the previous record won the EFF award should be retained in my opinion. -- Toshio Yamaguchi 18:18, 5 February 2013 (UTC)

I see now that PrimeHunter restored it. -- Toshio Yamaguchi 18:20, 5 February 2013 (UTC)

Are all smaller primes known?

What is the largest known prime number for which we have confirmed the prime or composite nature of every smaller integer? Is it the same as the current record for largest prime (M57885161) or are there definitely "gaps" of uncertainty? 23.30.218.174 (talk) 17:47, 17 April 2013 (UTC)

I guess it would be the largest prime number below the smallest probable prime whose primality hasn't been verified so far. The smallest PRP listed at http://www.primenumbers.net/prptop/prptop.php?page=40#haut is 2⁹⁰⁶⁹⁰-2²⁵-1, but since it only lists the top 10000, there might be PRPs whose primality hasn't been proven below that. If that is the case, then I don't know whether there exists a source that tracks the status of those lower numbers. List of prime numbers at the bottom contains lists of successive primes up to some specific bounds. There is a link to a list of all primes up to 1,000,000,000,000, which I cannot access because the link seems to be broken. http://www.prime-numbers.org/ contains a complete listing of all primes up to 10¹⁰ and a list up to 10³⁹⁹ seems to be available for cash. I think it would be possible to write a program for confirming the prime or composite status for all numbers below a bound higher than that in a reasonable time. -- Toshio Yamaguchi 18:10, 17 April 2013 (UTC)

Hmm, I had hoped I could compute many primes with PARI using the primes(x) command, but this works only up to ~10⁴. -- Toshio Yamaguchi 18:47, 17 April 2013 (UTC)

The above is wrong. Per the prime number theorem there are around 10³⁹⁶ primes below 10³⁹⁹. If every atom in the observable universe was a supercomputer which could compute billions of primes per nanosecond then the time since the Big Bang wouldn't be enough to reach 10¹²⁰. Small primes are so easy to compute and so numerous that nobody bothers to store the billions which can be computed in minutes on a PC. The primes are just recomputed when they are wanted for something. Depending on hardware and software, they may be faster to recompute than read from a harddisk.[4] I don't know the largest collection of stored primes but it might also be a candidate for the largest waste of storage space. The Goldbach conjecture verification project [5] has computed all primes below 4×10¹⁸. That's 95676260903887607 primes (near 10¹⁷).[6] They were not stored. If a compressed format with a byte per prime had been used then it would have required nearly one hundred thousand 1000 GB harddisks. As far as I know, 4×10¹⁸ (or slightly above that) is the largest number for which all smaller primes have been computed at some time. It would only take a small fraction of a second to compute the next many primes. For example, this naive PARI/GP line took 125 ms on my PC:

for(n=0,1000,if(isprime(4*10^18+n),print1(n", ")))

37, 49, 69, 121, 147, 163, 169, 267, 273, 333, 351, 393, 487, 489, 511, 567, 613, 639, 651, 679, 687, 727, 729, 771, 781, 793, 831, 903, 951, 979,

PrimeHunter (talk) 23:49, 17 April 2013 (UTC)

Well, the site http://www.prime-numbers.org/premium.html claims that you can download a list containing all primes up to 10⁴⁰⁰ for 19.99$, but I don't know whether anyone has bought and tried to download that file. -- Toshio Yamaguchi 08:14, 18 April 2013 (UTC)

No, http://www.prime-numbers.org/premium.html says: "All lists are partial list except the first list[0-10¹⁰), for partial list, the prime numbers are collected randomly from the full range." It also says how many primes in each range are included. PrimeHunter (talk) 12:30, 18 April 2013 (UTC)

It seems that you are correct and the last offer contains all previous zip files instead of a large one. -- Toshio Yamaguchi 13:05, 18 April 2013 (UTC)

Also, what does confirm mean anyway. I mean, one could write a program that primality tests each prime below a given upper bound without storing the primes. It would mean the prime or composite status of each number below that bound would have been confirmed by the computer. -- Toshio Yamaguchi 08:21, 18 April 2013 (UTC)

Thanks for all these answers! I had considered the "rewriting" point and am glad to see that my intuition is close to correct (even if each prime were "stored" as a single bit, and each bit occupied a single atom, there still wouldn't be enough atoms in the universe for all the smaller-than-largest-known primes... or something like that).

In a way, it's a philosophical question: How would we describe the process of a computer that continually overwrites currently stored primes with new ones? We might call it "confirming" but if the primes are being erased than what is being confirmed? Anyway, fascinating stuff. Yay primes! ± Lenoxus (" *** ") 23:01, 18 April 2013 (UTC)

Number format

In history, periods are used instead of commas. Am I recommended to use commas instead of periods? --Kc kennylau (talk) 09:12, 20 July 2013 (UTC)

Yes, but something funny is going on, the source does not seem to actually contain any decimal separators.—Emil J. 21:55, 20 July 2013 (UTC)

M₂₁₆₀₉₁: found in 1984 or 1985?

According to Graham, Knuth & Patashnik (1994), p. 109, M₂₁₆₀₉₁ was found in 1984 not in 1985:

Computer scientists at Chevron Geosciences did, however, strike mathematical oil in 1984. Using a program developed by David Slowinski, they discovered the largest prime known at that time, $2^{216091}-1$ while testing a new Cray X-MP supercomputer.

Helder 12:16, 14 December 2013 (UTC)

That's a single source disagreeing with almost all others. The Google search Slowinski 216091 (1984 OR 1985) is completely dominated by 1985. PrimeHunter (talk) 20:37, 14 December 2013 (UTC)

Question?

I found out that 10ⁿ +1 is a prime number, as n approaches infinity. Also, any number ending in -59 is prime too. — Preceding unsigned comment added by Johndric Valdez (talk • contribs) 07:12, 8 March 2014 (UTC)

Also, I've found out that 2⁴ⁿ +1 is prime, as n approaches infinity. Johndric Valdez (talk) 07:15, 8 March 2014 (UTC)

Seems very unlikely to me considering the prime number theorem. How did you 'find out'? -- Toshio Yamaguchi 09:41, 8 March 2014 (UTC)

The claims are all nonsense. If "as n approaches infinity" is supposed to mean "for large n" (the sequences diverge), the claims are known to be false: 2⁴ⁿ +1 can only be prime for 4n a power of 2, see Fermat number. And 10ⁿ +1 contains infinitely many numbers of the form n^n+1 (namely n=10^k for all k), which can only be prime for n a power of 2.[7] Finally, the "-59" claim is a blatant contradiction to the prime number theorem. --Roentgenium111 (talk) 15:12, 30 June 2014 (UTC)

Not only is it in blatant contradiction to the prime number theorem, it also fails already for the second number it could be applied to, as 159 = 3 × 53. Double sharp (talk) 13:26, 20 March 2015 (UTC)

A few early records

I added a few early records from [8]. Some of them are disputed; I've included explanatory notes.

(Given that Fermat apparently had the tools to do so, according to Fermat number, it's really a shame that he didn't factor F₅ = 2³² + 1 = 4,294,967,297 = 641 × 6,700,417, because 6,700,417 would've been a record at 7 digits in the 17th century!) Double sharp (talk) 13:31, 20 March 2015 (UTC)

P.S. If anyone remembers Euler's factorization of F₅; yes, Euler did show that 641 divides F₅ in 1732, but annoyingly he never seems to have mentioned its other prime factor 6,700,417, much less its primality, in that paper! So it may not be a record, and I haven't found anyone including it. :-( Double sharp (talk) 13:35, 20 March 2015 (UTC)

Hmm. Fermat number#Factorization of Fermat numbers does give 6,700,417 as being found as a factor in 1732 by Euler. Maybe it's not in the same paper as the one he finds 641 in? Adding it in, then. Double sharp (talk) 13:42, 20 March 2015 (UTC)

Date confusion again

The article states that

As of October 2015, the largest known prime number is 257,885,161 − 1, a number with 17,425,170 digits. It was found in 2013 by the Great Internet Mersenne Prime Search (GIMPS).

Which date is the correct one and is the methodology still correct? — Preceding unsigned comment added by Mbsaerens (talk • contribs) 16:23, 19 November 2015 (UTC)

@Mbsaerens: It means that the number was found in 2013 and is still the largest known prime as of October 2015. In the future, please create new talk page sections at the bottom, see WP:TALK. -- intgr [talk] 20:16, 19 November 2015 (UTC)

Yes, see Wikipedia:As of for the recommend method add to such statements, which is followed here. PrimeHunter (talk) 20:41, 19 November 2015 (UTC)

new prime found

http://www.mersenne.org/
http://primes.utm.edu/largest.html
2^274,207,281-1 with 22,338,618 digits
07 jan 2016 — Preceding unsigned comment added by 82.1.144.11 (talk) 14:19, 19 January 2016 (UTC)

this is useless, even the number itself is wrong (where that extra 2 came from?) Trimutius (talk) 16:00, 19 January 2016 (UTC)

Stupidly early records

The earliest possible record may well be 23, which appears in the Rhind Papyrus, receiving a special mark together with 3, 5, 7, and 11. After that, the most likely candidate may be 127 from Euclid's Elements. But all this is very dodgy and I would not include it. Double sharp (talk) 15:54, 18 April 2016 (UTC)

Well I wouldn't say that early records can be a reliable source of highest known primes. I doubt that Euclid didn't go higher than 127. It might have been largest known Mersenne Prime for a while, but Mersenne Primes are not the only ones. Trimutius (talk) 18:34, 18 April 2016 (UTC)

I am confused by the first two entries in the history list of largest known (demonstrated?) primes. To wit, first comes eleven, then comes seven. How can this be? 11>7, and the date for 11 is earlier. Is there a seventh columnist here -- or am I missing prime factor?Kdammers (talk) 03:19, 8 January 2018 (UTC)

The notes column for 11 says "disputed" and has an inline reference with an explanation. But the whole idea of a tiny number like 7, 11 or 127 being the largest known prime at some time is problematic. PrimeHunter (talk) 11:47, 8 January 2018 (UTC)

I removed the 3 tiny "records". Those tiny numbers (7, 11, 127) can not possibly have been "largest know". What "largest known" was, at those times in history, is impossible to find out. But the largest knowns MUST have been greater than those tiny ones stated. -- DexterPointy (talk) 20:30, 22 February 2018 (UTC)

I totally agree with removing 7 and 11. I'm not quite sure about 127, which could have conceivably been the largest Euclid bothered to check, since the Sieve of Eratosthenes was not yet available. Meanwhile, the next two primes listed (M_13 and M_17) are also dubious according to the given Caldwell source [9] which attributes the latter to Cataldi (1588, simultaneously with M_19) and disregards 15th century claims for M_13 and M_17 "for lack of evidence that they were proven primes at that time, rather than just lucky guesses." (Strangely, Caldwell does accept an anonymous 1456 discovery for M_13 in his Mersenne primes list [10]) Roentgenium111 (talk) 16:56, 24 February 2018 (UTC)

That Euclid might not have had any particular interest in finding large primes, could easily be true. But that he somehow never "accidentally" encountered quite a few numbers greater than 127, and found a greater prime by "accidentally" checking for primality, is totally unbelievable. (Euclid wasn't exactly a retarded child of the pocket-calculator age). Some of the lost works by Euclid, includes some on mechanics, and that include theory of leavers. Ratio & Division is about as close related as can be, and a cornerstone in studying leavers. (Also, let's not forget that 60, as in Sexagesimal, is remarkably old, and likely anything but a random choice, and is also used as radix in ancient greek trigonometric tables.) --DexterPointy (talk) 21:36, 24 February 2018 (UTC)

Largest Prime Number Found "by Hand / Not by Computer"

The Mersenne prime article states:

Édouard Lucas proved in 1876 that M127 is indeed prime, as Mersenne claimed. This was the largest known prime number for 75 years, and the largest ever found by hand.

However, this article states, in the table, that ${\tfrac {2^{148}+1}{17}}$ was "Found by Aimé Ferrier; the largest record not set by computer".

This superficially appears to be a contradiction. I believe the resolution is that M127 was found by hand; and Ferrier's number was found using a mechanical calculator (so not by computer). See numericana.com.

So I guess the article could clarify this, and possibly state the specific rationale for both records. -- johantheghost (talk) 09:49, 14 May 2016 (UTC)

Prime ending in 2

The largest prime as of Jan 2018 (2^77,232,917 − 1) is stated to end in '2'. I think that stops it being a prime number. What am I missing? Stratopastor (talk) 00:05, 6 January 2018 (UTC)

@Stratopastor: You are right. An IP changed the last digit from '1' to '2'. It has been fixed.[11] PrimeHunter (talk) 03:02, 6 January 2018 (UTC)

Thanks @PrimeHunter ! What's an IP in this context? Stratopastor (talk) 19:00, 6 January 2018 (UTC)

@Stratopastor: An IP means an editor who is not logged in and automatically has their IP address listed at edits instead of a username. The error was made by the IP 129.67.127.13 and fixed by another IP 190.189.92.14. PrimeHunter (talk) 19:17, 6 January 2018 (UTC)