Largest known prime number
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. The last 15 record primes were Mersenne primes.
The current record
The record is currently held by 257,885,161 − 1 with 17,425,170 digits. The value of it is
- ... (17,425,042 digits omitted) ...
The first and last 64 digits are shown above, with the intervening digits omitted.
Its discovery resulted from the Great Internet Mersenne Prime Search (GIMPS).
The record passed one million digits in 1999, earning a $50,000 prize. In 2008 the record passed ten million digits, earning a $100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation. Time called it the 29th top invention of 2008. 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.
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. The longest record-holder known was M19 = 524,287, which was the largest known prime for 144 years. No records are known before 1456.
|Number||# digits||Year found||Notes|
|M13||8,191||4||1456||Anonymous discovery: no proof provided, so sometimes not counted as a record.|
|M17||131,071||6||1460||Anonymous discovery: no proof provided. Later proven prime by Pietro Cataldi in 1588, and that is sometimes listed as a record.|
|M19||524,287||6||1588||Found by Pietro Cataldi|
|(232+1)/641||6,700,417||7||1732||Found by Leonhard Euler|
|M31||2,147,483,647||10||1772||Found by Leonhard Euler|
|(1018+1)/1000001||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.|
|(264+1)/274177||67,280,421,310,721||14||1855||Found by Thomas Clausen, but proof was not provided: thus this record is not always listed.|
|[M59/179951]||[3,203,431,780,337]||||||Found by Landry. A record if the immediately preceding entry is excluded.|
|M127||39||1876||Found by Édouard Lucas|
|(2148+1)/17||44||1951||Found by Ferrier; the largest record not set by computer.|
|180×(M127)2+1||79||1951||Using Cambridge's EDSAC computer|
The ten largest known prime numbers
|Rank||Prime number||Found by||Found date||Number of digits||Reference|
|1st||257,885,161 − 1||GIMPS||2013 January 25||17,425,170|||
|2nd||243,112,609 − 1||GIMPS||2008 August 23||12,978,189|||
|3rd||242,643,801 − 1||GIMPS||2009 April 12||12,837,064|||
|4th||237,156,667 − 1||GIMPS||2008 September 6||11,185,272|||
|5th||232,582,657 − 1||GIMPS||2006 September 4||9,808,358|||
|6th||230,402,457 − 1||GIMPS||2005 December 15||9,152,052|||
|7th||225,964,951 − 1||GIMPS||2005 February 18||7,816,230|||
|8th||224,036,583 − 1||GIMPS||2004 May 15||7,235,733|||
|9th||220,996,011 − 1||GIMPS||2003 November 17||6,320,430|||
|10th||213,466,917 − 1||GIMPS||2001 November 14||4,053,946|||
GIMPS found the eleven latest records on ordinary computers operated by participants around the world.
- Mersenne prime
- Primality test
- Prime number
- Fermat number#Generalized Fermat primes
- Cullen number
- Sophie Germain prime
- Woodall number
- "GIMPS Project Discovers Largest Known Prime Number, 257,885,161-1". Mersenne Research, Inc.
- Chris Caldwell, The largest known primes. Retrieved on 2013-02-05.
- "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. Retrieved January 17, 2012.
- Landon Curt Noll, Mersenne Prime Digits and Names. Retrieved on 2011-01-03.
- Samuel Yates, Chris Caldwell, The largest known primes. Retrieved on 2014-03-08.
- "Press release, 5 May 2007". Seventeen or Bust.
- 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