Largest known prime number
The largest known prime number is 257,885,161 − 1[1], a number with 17,425,170 digits.
It was proven by Euclid that there are infinitely many prime numbers. Thus, there are always infinitely many primes greater than the largest known prime. Many mathematicians and hobbyists search for large prime numbers. There are several prizes offered by the Electronic Frontier Foundation for record primes.[2]
The fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is fast compared to other known primality tests for other kinds of numbers. Due in part to this and to the historical interest in Mersenne primes, many of the largest known primes are Mersenne primes. As of February 2013[update] the ten largest known primes are Mersenne primes, while the eleventh is the largest known non-Mersenne prime.[3] The last 15 record primes were Mersenne primes. Before that was a single non-Mersenne (improving the record by merely 37 digits in 1989), and 17 more Mersenne primes going back to 1952.[4]
The use of electronic computers has accelerated the discoveries and found all records since 1951. The record passed one million digits in 1999, earning a $50,000 prize.[5] In 2008 the record passed ten million digits, earning a $100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation.[2] Time called it the 29th top invention of 2008.[6] Additional prizes are being offered for the first prime number found with at least one hundred million digits and the first with at least one billion digits.[2]
| Rank | Prime number | Found by | Found date | Number of digits | Reference |
|---|---|---|---|---|---|
| 1st | 257,885,161 − 1 | GIMPS | 2013 January 25 | 17,425,170 | [3] |
| 2nd | 243,112,609 − 1 | GIMPS | 2008 August 23 | 12,978,189 | [3] |
| 3rd | 242,643,801 − 1 | GIMPS | 2009 April 12 | 12,837,064 | [7] |
| 4th | 237,156,667 − 1 | GIMPS | 2008 September 6 | 11,185,272 | [7] |
| 5th | 232,582,657 − 1 | GIMPS | 2006 September 4 | 9,808,358 | [7] |
| 6th | 230,402,457 − 1 | GIMPS | 2005 December 15 | 9,152,052 | [8] |
| 7th | 225,964,951 − 1 | GIMPS | 2005 February 18 | 7,816,230 | [8] |
| 8th | 224,036,583 − 1 | GIMPS | 2004 May 15 | 7,235,733 | [8] |
| 9th | 220,996,011 − 1 | GIMPS | 2003 November 17 | 6,320,430 | [8] |
| 10th | 213,466,917 − 1 | GIMPS | 2001 November 14 | 4,053,946 | [8] |
GIMPS found the 11 latest records on ordinary computers operated by participants around the world.
Contents |
The current record [edit]
The record is currently held by 257,885,161 − 1 with 17,425,170 digits. Its discovery resulted from the Great Internet Mersenne Prime Search (GIMPS).
History [edit]
The following table lists the progression of the largest known prime number in ascending order. Here Mn= 2n − 1 is the Mersenne number with exponent n.[4]
| Number | Digits | Year found |
|---|---|---|
| M127 | 39 | 1876 |
| 180×(M127)2 + 1 | 79 | 1951 |
| M521 | 157 | 1952 |
| M607 | 183 | 1952 |
| M1279 | 386 | 1952 |
| M2203 | 664 | 1952 |
| M2281 | 687 | 1952 |
| M3217 | 969 | 1957 |
| M4423 | 1.332 | 1961 |
| M9689 | 2.917 | 1963 |
| M9941 | 2.993 | 1963 |
| M11213 | 3.376 | 1963 |
| M19937 | 6.002 | 1971 |
| M21701 | 6.533 | 1978 |
| M23209 | 6.987 | 1979 |
| M44497 | 13.395 | 1979 |
| M86243 | 25.962 | 1982 |
| M132049 | 39.751 | 1983 |
| M216091 | 65.050 | 1985 |
| 391581×2216193 − 1 | 65.087 | 1989 |
| M756839 | 227.832 | 1992 |
| M859433 | 258.716 | 1994 |
| M1257787 | 378.632 | 1996 |
| M1398269 | 420.921 | 1996 |
| M2976221 | 895.932 | 1997 |
| M3021377 | 909.526 | 1998 |
| M6972593 | 2.098.960 | 1999 |
| M13466917 | 4.053.946 | 2001 |
| M20996011 | 6.320.430 | 2003 |
| M24036583 | 7.235.733 | 2004 |
| M25964951 | 7.816.230 | 2005 |
| M30402457 | 9.152.052 | 2005 |
| M32582657 | 9.808.358 | 2006 |
| M43112609 | 12.978.189 | 2008 |
| M57885161 | 17.425.170 | 2013 |
References [edit]
- ^ "GIMPS Project Discovers Largest Known Prime Number, 257,885,161-1". Mersenne Research, Inc.
- ^ 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.
- ^ a b c Chris Caldwell, The largest known primes. Retrieved on 2013-02-05.
- ^ a b Chris Caldwell, The largest known prime by year.
- ^ Electronic Frontier Foundation, Big Prime Nets Big Prize.
- ^ "Best Inventions of 2008 - 29. The 46th Mersenne Prime". Time. Time Inc. Retrieved January 17, 2012.
- ^ a b c Landon Curt Noll, Mersenne Prime Digits and Names. Retrieved on 2011-01-03.
- ^ a b c d e Samuel Yates, Chris Caldwell, The largest known primes. Retrieved on 2012-06-09.
External links [edit]
- Press release about the largest known prime 257,885,161−1
- Press release about the former largest known prime 243,112,609−1
- Press release about an earlier largest known prime 232,582,657−1
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