The largest known prime number is 282,589,933 − 1, a number which has 24,862,048 digits when written in base 10. It was found via a computer volunteered by Patrick Laroche of the Great Internet Mersenne Prime Search (GIMPS) in 2018.[1]
A prime number is a natural number greater than 1 with no divisors other than 1 and itself. According to Euclid's theorem there are infinitely many prime numbers, so there is no largest prime.
Many of the largest known primes are Mersenne primes, numbers that are one less than a power of two, because they can utilize a specialized primality test that is faster than the general one. As of June 2023[update], the six largest known primes are Mersenne primes.[2] The last seventeen record primes were Mersenne primes.[3][4] The binary representation of any Mersenne prime is composed of all ones, since the binary form of 2k − 1 is simply k ones.[5]
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Transcription
DR. TONY PADILLA: The GIMPS have been at it again. They've found a new prime number, the largest prime number that's been found to date. The GIMPS, of course, are the Great Internet Mersenne Prime Search team, a network of computers and people that work on these computers that look for large Mersenne primes. It's a pretty big number, 2 to the power-- now, I've got to remember this, now-- 57,885,161 minus 1. So the 57,885,161 is a prime number. But this bad boy, the whole thing, is also a prime number, of course. It's a much bigger one. And that is the largest prime number ever to have been found. It's the 48th Mersenne prime ever to have been found as well. BRADY HARAN: Can you remind us what a Mersenne prime is? DR. TONY PADILLA: Yes. So essentially, the Mersenne prime-- you just take 2. You take it to some power, P-- and this is a prime number-- minus 1. And that's your Mersenne prime. It's pretty interesting how they've gone about finding this, as well. There's various things about it. For example, the way it's done is they have this GIMPS network. They're a network of shared computers. And they use their combined computing power to basically work out, try and find these prime numbers. So it's quite interesting, this sort of collaboration going on to find these things. From my point of view, it's actually reminiscent of something that they do at the LHC to manage the data. They basically use a network of computers around the world to analyze that data. So it's not always academics that are finding these things. Anybody can get involved in the project. And there's money on the table, as well, when it comes to finding these things. Curtis Cooper will win-- it looks like $3,000 for finding this number. It's a bit of a shame for him because this number, actually-- well, it has this many digits. It's this many digits long. It is 17,425,170 digits long, I think. The reason this is a little bit annoying for Curtis Cooper is that it's not 100 million digits long. Because if it had been 100 million digits long, then he would have won $150,000. Because that's the prize on offer if he can find one that's actually that long. It's worth thinking about how big this number actually is. And we've talked about big numbers in the past, Brady. It is bigger than a googol. It's not as big as a googolplex. It's certainly not as big as Graham's number. You can easily think of this number without your head collapsing to form a black hole. There's no problem there. In fact, actually, it's quite easy to think of. You could even memorize this number. Here we have the number. So this is the number written out in full-- quite a big file. There you go. It's got 17 and 1/2 million digits. This file is about 22 meg. Now, apparently, your brain can store about 2.5 million gig of memory. So technically, you could memorize this number without having a problem. And in fact, I'm going to be cocky here because I actually can write this. I'm going to write this number. I've remembered this number. But I've remembered it in base 2. OK. So let me write down it out in base 2. I was quite impressed that I remembered this. So the first number is 1. The second number-- ah, it's also 1. And the next one's also 1. And the next one is also 1. And I would carry on writing 1's until I got all the way to the last number, which is also 1. And there would be a grand total of 57,885,161 of these 1's. And that is the number written in base 2. So I can prove that for you if you'd like. BRADY HARAN: Go on. DR. TONY PADILLA: OK. So the reason that that's what it is in base 2-- OK. So let's look at this number. Let's just call it P for now, rather than this 57 million. Because we can prove this in generality. I can actually factorize this expression. And I can factorize this expression as follows. It's 2 minus 1. That's obviously 1-- multiplied by 1, plus 2, plus 2 squared, all the way up to 2 P minus 1. OK. So this is obviously 1 out front. So what I'm left with is just all the, basically, essential units of base 2 with 1's in front. And that's it. So that's the proof. So there's obviously P of these in total. So in our case, P is 57,885,161. And so this is how you would write it out in base 2. BRADY HARAN: In base 10, which is what more people are familiar with, what's the first and last digit? DR. TONY PADILLA: So I'm going to have to look this one up. Well, I think I know the first, actually. actually. It's 5, isn't it? It's 5. And the last one-- can I look? BRADY HARAN: Yeah. DR. TONY PADILLA: I've got a feeling the last one's-- well, we know it's going to be an odd number, don't we? So we've got 1. There we go. Of course, we have a new perfect number from this. And it's very easy to write down what it is. And the perfect number that we have-- 2 to the 57,885,160, multiplied by 2 to the 57,885,161 minus 1. That's our new perfect number. It's about 34 million digits long. Yep. That's our new perfect number we've got out of this. So whew. It's all good fun, basically. And I think that there is an important point to come from-- these searches. I mean, they're fun to look at. It's nice. It's interesting. Oh, we found this new, big prime number. There's a lot of technology that goes into sort of developing the software to do these calculations. Because basically, if you actually tried to calculate all the divisors of this guy, you've used all the numbers that were smaller than it and tried to see if they were factors of that number, if you did that, just like that, sort of blindly, you'd find that you needed more than the age of the universe to actually do that calculation. So there's sophisticated techniques at work here. And all this game, and all this developing this computing power in particular ways to multiply large numbers, really have had sort of impact elsewhere, for example. It's the spinoff technologies that are coming from this sort of really mathematicians playing games. But you should never knock mathematicians playing games. Because the crumbs that fall off the table can be of benefit to everybody else. BRADY HARAN: If you'd like to see even more from this interview, including a rather strange discussion about my family tree, I've got an unlisted video with all the off cuts and extra footage. I basically didn't put it as a listed video because I didn't want to clog up everyone's subscription boxes. And if you want to find out more about some of these topics and you haven't watched our back catalogue of videos, I've also put links to our films about Mersenne primes, perfect numbers, googols and googolplexes, Graham's number, all that kind of stuff. Check out the links under the video or here on the screen.
Current record
The record is currently held by 282,589,933 − 1 with 24,862,048 digits, found by GIMPS in December 2018.[1] The first and last 120 digits of its value are shown below:
148894445742041325547806458472397916603026273992795324185271289425213239361064475310309971132180337174752834401423587560 ...
(24,861,808 digits skipped)
... 062107557947958297531595208807192693676521782184472526640076912114355308311969487633766457823695074037951210325217902591[6]
As of February 2024[update], this prime has held the record for more than 5 years, longer than any other prime since M19937 (which held the record for 7 years from 1971 to 1978).
Prizes
There are several prizes offered by the Electronic Frontier Foundation (EFF) for record primes.[7] A prime with one million digits was found in 1999, earning the discoverer a US$50,000 prize.[8] In 2008, a ten-million digit prime won a US$100,000 prize and a Cooperative Computing Award from the EFF.[7] Time called this prime the 29th top invention of 2008.[9]
Both of these primes were discovered through the Great Internet Mersenne Prime Search (GIMPS), which coordinates long-range search efforts among tens of thousands of computers and thousands of volunteers. The $50,000 prize went to the discoverer and the $100,000 prize went to GIMPS. GIMPS will split the US$150,000 prize for the first prime of over 100 million digits with the winning participant. A further prize is offered for the first prime with at least one billion digits.[7]
GIMPS also offers a US$3,000 research discovery award for participants who discover a new Mersenne prime of less than 100 million digits.[10]
History of largest known prime numbers
The following table lists the progression of the largest known prime number in ascending order.[3] Here Mp = 2p − 1 is the Mersenne number with exponent p, where p is a prime number. The longest record-holder known was M19 = 524,287, which was the largest known prime for 144 years. No records are known prior to 1456.
Number | Decimal expansion (partial for numbers > M1000) |
Digits | Year found | Discoverer |
---|---|---|---|---|
M13 | 8,191 | 4 | 1456 | Anonymous |
M17 | 131,071 | 6 | 1588 | Pietro Cataldi |
M19 | 524,287 | 6 | 1588 | Pietro Cataldi |
6,700,417 | 7 | 1732 | Leonhard Euler? Euler did not explicitly publish the primality of 6,700,417, but the techniques he had used to factorise 232 + 1 meant that he had already done most of the work needed to prove this, and some experts believe he knew of it.[11] | |
M31 | 2,147,483,647 | 10 | 1772 | Leonhard Euler |
999,999,000,001 | 12 | 1851 | Included (but question-marked) in a list of primes by Looff. Given his uncertainty, some do not include this as a record. | |
67,280,421,310,721 | 14 | 1855 | Thomas Clausen (but no proof was provided). | |
M127 | 170,141,183,460,469, |
39 | 1876 | Édouard Lucas |
20,988,936,657,440, |
44 | 1951 | Aimé Ferrier with a mechanical calculator; the largest record not set by computer. | |
180×(M127)2+1 |
521064401567922879406069432539 |
79 | 1951 | J. C. P. Miller & D. J. Wheeler[12] Using Cambridge's EDSAC computer |
M521 |
686479766013060971498190079908 |
157 | 1952 | Raphael M. Robinson |
M607 |
531137992816767098689588206552 |
183 | 1952 | Raphael M. Robinson |
M1279 | 104079321946...703168729087 | 386 | 1952 | Raphael M. Robinson |
M2203 | 147597991521...686697771007 | 664 | 1952 | Raphael M. Robinson |
M2281 | 446087557183...418132836351 | 687 | 1952 | Raphael M. Robinson |
M3217 | 259117086013...362909315071 | 969 | 1957 | Hans Riesel |
M4423 | 285542542228...902608580607 | 1,332 | 1961 | Alexander Hurwitz |
M9689 | 478220278805...826225754111 | 2,917 | 1963 | Donald B. Gillies |
M9941 | 346088282490...883789463551 | 2,993 | 1963 | Donald B. Gillies |
M11213 | 281411201369...087696392191 | 3,376 | 1963 | Donald B. Gillies |
M19937 | 431542479738...030968041471 | 6,002 | 1971 | Bryant Tuckerman |
M21701 | 448679166119...353511882751 | 6,533 | 1978 | Laura A. Nickel and Landon Curt Noll[13] |
M23209 | 402874115778...523779264511 | 6,987 | 1979 | Landon Curt Noll[13] |
M44497 | 854509824303...961011228671 | 13,395 | 1979 | David Slowinski and Harry L. Nelson[13] |
M86243 | 536927995502...709433438207 | 25,962 | 1982 | David Slowinski[13] |
M132049 | 512740276269...455730061311 | 39,751 | 1983 | David Slowinski[13] |
M216091 | 746093103064...103815528447 | 65,050 | 1985 | David Slowinski[13] |
148140632376...836387377151 | 65,087 | 1989 | A group, "Amdahl Six": John Brown, Landon Curt Noll, B. K. Parady, Gene Ward Smith, Joel F. Smith, Sergio E. Zarantonello.[14][15] Largest non-Mersenne prime that was the largest known prime when it was discovered. | |
M756839 | 174135906820...328544677887 | 227,832 | 1992 | David Slowinski and Paul Gage[13] |
M859433 | 129498125604...243500142591 | 258,716 | 1994 | David Slowinski and Paul Gage[13] |
M1257787 | 412245773621...976089366527 | 378,632 | 1996 | David Slowinski and Paul Gage[13] |
M1398269 | 814717564412...868451315711 | 420,921 | 1996 | GIMPS, Joel Armengaud |
M2976221 | 623340076248...743729201151 | 895,932 | 1997 | GIMPS, Gordon Spence |
M3021377 | 127411683030...973024694271 | 909,526 | 1998 | GIMPS, Roland Clarkson |
M6972593 | 437075744127...142924193791 | 2,098,960 | 1999 | GIMPS, Nayan Hajratwala |
M13466917 | 924947738006...470256259071 | 4,053,946 | 2001 | GIMPS, Michael Cameron |
M20996011 | 125976895450...762855682047 | 6,320,430 | 2003 | GIMPS, Michael Shafer |
M24036583 | 299410429404...882733969407 | 7,235,733 | 2004 | GIMPS, Josh Findley |
M25964951 | 122164630061...280577077247 | 7,816,230 | 2005 | GIMPS, Martin Nowak |
M30402457 | 315416475618...411652943871 | 9,152,052 | 2005 | GIMPS, University of Central Missouri professors Curtis Cooper and Steven Boone |
M32582657 | 124575026015...154053967871 | 9,808,358 | 2006 | GIMPS, Curtis Cooper and Steven Boone |
M43112609 | 316470269330...166697152511 | 12,978,189 | 2008 | GIMPS, Edson Smith |
M57885161 | 581887266232...071724285951 | 17,425,170 | 2013 | GIMPS, Curtis Cooper |
M74207281 | 300376418084...391086436351 | 22,338,618 | 2016 | GIMPS, Curtis Cooper |
M77232917 | 467333183359...069762179071 | 23,249,425 | 2017 | GIMPS, Jonathan Pace |
M82589933 | 148894445742...325217902591 | 24,862,048 | 2018 | GIMPS, Patrick Laroche |
GIMPS found the fifteen latest records (all of them Mersenne primes) on ordinary computers operated by participants around the world.
The twenty largest known prime numbers
A list of the 5,000 largest known primes is maintained by the PrimePages,[16] of which the twenty largest are listed below.[17]
Rank | Number | Discovered | Digits | Form | Ref |
---|---|---|---|---|---|
1 | 282589933 − 1 | 2018-12-07 | 24,862,048 | Mersenne | [1] |
2 | 277232917 − 1 | 2017-12-26 | 23,249,425 | Mersenne | [18] |
3 | 274207281 − 1 | 2016-01-07 | 22,338,618 | Mersenne | [19] |
4 | 257885161 − 1 | 2013-01-25 | 17,425,170 | Mersenne | [20] |
5 | 243112609 − 1 | 2008-08-23 | 12,978,189 | Mersenne | [21] |
6 | 242643801 − 1 | 2009-06-04 | 12,837,064 | Mersenne | [22] |
7 | Φ3(−5166931048576) | 2023-10-02 | 11,981,518 | Generalized unique | [23] |
8 | Φ3(−4658591048576) | 2023-05-31 | 11,887,192 | Generalized unique | [24] |
9 | 237156667 − 1 | 2008-09-06 | 11,185,272 | Mersenne | [21] |
10 | 232582657 − 1 | 2006-09-04 | 9,808,358 | Mersenne | [25] |
11 | 10223 × 231172165 + 1 | 2016-10-31 | 9,383,761 | Proth | [26] |
12 | 230402457 − 1 | 2005-12-15 | 9,152,052 | Mersenne | [27] |
13 | 225964951 − 1 | 2005-02-18 | 7,816,230 | Mersenne | [28] |
14 | 224036583 − 1 | 2004-05-15 | 7,235,733 | Mersenne | [29] |
15 | 19637361048576 + 1 | 2022-09-24 | 6,598,776 | Generalized Fermat | [30] |
16 | 19517341048576 + 1 | 2022-08-09 | 6,595,985 | Generalized Fermat | [31] |
17 | 202705 × 221320516 + 1 | 2021-11-25 | 6,418,121 | Proth | [32] |
18 | 220996011 − 1 | 2003-11-17 | 6,320,430 | Mersenne | [33] |
19 | 10590941048576 + 1 | 2018-10-31 | 6,317,602 | Generalized Fermat | [34] |
20 | 3 × 220928756 − 1 | 2023-07-05 | 6,300,184 | Thabit | [35] |
See also
References
- ^ a b c "GIMPS Project Discovers Largest Known Prime Number: 282,589,933-1". Mersenne Research, Inc. 21 December 2018. Retrieved 21 December 2018.
- ^ "The largest known primes – Database Search Output". Prime Pages. Retrieved 19 March 2023.
- ^ a b Caldwell, Chris. "The Largest Known Prime by Year: A Brief History". Prime Pages. Retrieved 19 March 2023.
- ^ The last non-Mersenne to be the largest known prime, was 391,581 ⋅ 2216,193 − 1; see also The Largest Known Prime by year: A Brief History originally by Caldwell.
- ^ "Perfect Numbers". Penn State University. Retrieved 6 October 2019.
An interesting side note is about the binary representations of those numbers...
- ^ "51st Known Mersenne Prime Discovered".
- ^ a b c "Record 12-Million-Digit Prime Number Nets $100,000 Prize". Electronic Frontier Foundation. Electronic Frontier Foundation. October 14, 2009. Retrieved November 26, 2011.
- ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
- ^ "Best Inventions of 2008 - 29. The 46th Mersenne Prime". Time. Time Inc. October 29, 2008. Archived from the original on November 2, 2008. Retrieved January 17, 2012.
- ^ "GIMPS by Mersenne Research, Inc". mersenne.org. Retrieved 21 November 2022.
- ^ Edward Sandifer, C. (19 November 2014). How Euler Did Even More. The Mathematical Association of America. ISBN 9780883855843.
- ^ J. Miller, Large Prime Numbers. Nature 168, 838 (1951).
- ^ a b c d e f g h i Landon Curt Noll, Large Prime Number Found by SGI/Cray Supercomputer.
- ^ Letters to the Editor. The American Mathematical Monthly 97, no. 3 (1990), p. 214. Accessed May 22, 2020.
- ^ Proof-code: Z, The Prime Pages.
- ^ "The Prime Database: The List of Largest Known Primes Home Page". t5k.org/primes. Retrieved 19 March 2023.
- ^ "The Top Twenty: Largest Known Primes". Retrieved 19 March 2023.
- ^ "GIMPS Project Discovers Largest Known Prime Number: 277,232,917-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 3 January 2018.
- ^ "GIMPS Project Discovers Largest Known Prime Number: 274,207,281-1". mersenne.org. Great Internet Mersenne Prime Search. Retrieved 29 September 2017.
- ^ "GIMPS Discovers 48th Mersenne Prime, 257,885,161-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 5 February 2013. Retrieved 29 September 2017.
- ^ a b "GIMPS Discovers 45th and 46th Mersenne Primes, 243,112,609-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 15 September 2008. Retrieved 29 September 2017.
- ^ "GIMPS Discovers 47th Mersenne Prime, 242,643,801-1 is newest, but not the largest, known Mersenne Prime". mersenne.org. Great Internet Mersenne Prime Search. 12 April 2009. Retrieved 29 September 2017.
- ^ "PrimePage Primes: Phi(3, - 516693^1048576)". t5k.org.
- ^ "PrimePage Primes: Phi(3, - 465859^1048576)". t5k.org.
- ^ "GIMPS Discovers 44th Mersenne Prime, 232,582,657-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 11 September 2006. Retrieved 29 September 2017.
- ^ "PrimeGrid's Seventeen or Bust Subproject" (PDF). primegrid.com. PrimeGrid. Retrieved 30 September 2017.
- ^ "GIMPS Discovers 43rd Mersenne Prime, 230,402,457-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 24 December 2005. Retrieved 29 September 2017.
- ^ "GIMPS Discovers 42nd Mersenne Prime, 225,964,951-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 27 February 2005. Retrieved 29 September 2017.
- ^ "GIMPS Discovers 41st Mersenne Prime, 224,036,583-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 28 May 2004. Retrieved 29 September 2017.
- ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 October 2022.
- ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 17 September 2022.
- ^ "PrimeGrid's Extended Sierpinski Problem Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 28 December 2021.
- ^ "GIMPS Discovers 40th Mersenne Prime, 220,996,011-1 is now the Largest Known Prime". mersenne.org. Great Internet Mersenne Prime Search. 2 December 2003. Retrieved 29 September 2017.
- ^ "PrimeGrid's Generalized Fermat Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 7 November 2018.
- ^ "PrimeGrid's 321 Prime Search" (PDF). primegrid.com. PrimeGrid. Retrieved 17 July 2023.
External links
- Press release about the largest known prime 282,589,933−1
- Press release about the former largest known prime 277,232,917−1
- Press release about the former largest known prime 274,207,281−1