Largest known prime
Encyclopedia
The largest known prime number is the largest integer
that is currently known to be a prime number
.
It was proven
by Euclid
that there are infinitely many prime numbers; thus, there is always a prime 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.
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. the nine largest known primes are Mersenne primes, while the tenth is the largest known non-Mersenne prime. The last 14 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.
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. In 2008 the record passed ten million digits, earning a $100,000 prize. 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.
GIMPS found the 11 latest records on ordinary computers operated by participants around the world.
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
that is currently known to be a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
.
It was proven
Euclid's theorem
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem.-Euclid's proof:...
by Euclid
Euclid
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
that there are infinitely many prime numbers; thus, there is always a prime 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
Electronic Frontier Foundation
The Electronic Frontier Foundation is an international non-profit digital rights advocacy and legal organization based in the United States...
for record primes.
The fast Fourier transform
Fast Fourier transform
A fast Fourier transform is an efficient algorithm to compute the discrete Fourier transform and its inverse. "The FFT has been called the most important numerical algorithm of our lifetime ." There are many distinct FFT algorithms involving a wide range of mathematics, from simple...
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. the nine largest known primes are Mersenne primes, while the tenth is the largest known non-Mersenne prime. The last 14 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.
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. In 2008 the record passed ten million digits, earning a $100,000 prize. 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.
| Rank | Prime number | Found by | Found date | Number of digits | Reference |
|---|---|---|---|---|---|
| 1st | 243,112,609 − 1 | GIMPS Great Internet Mersenne Prime Search The Great Internet Mersenne Prime Search is a collaborative project of volunteers who use freely available computer software to search for Mersenne prime numbers. The project was founded by George Woltman, who also wrote the software Prime95 and MPrime for the project... |
23 August 2008 | 12,978,189 | |
| 2nd | 242,643,801 − 1 | GIMPS | April 2009 | 12,837,064 | |
| 3rd | 237,156,667 − 1 | GIMPS | 6 September 2008 | 11,185,272 | |
| 4th | 232,582,657 − 1 | GIMPS | 4 September 2006 | 9,808,358 |
GIMPS found the 11 latest records on ordinary computers operated by participants around the world.
History
The following table lists the progression of the largest known prime number in ascending order, where Mn is a Mersenne number with exponent n.| 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 | 1332 | 1961 |
| M9689 | 2917 | 1963 |
| M9941 | 2993 | 1963 |
| M11213 | 3376 | 1963 |
| M19937 | 6002 | 1971 |
| M21701 | 6533 | 1978 |
| M23209 | 6987 | 1979 |
| M44497 | 13395 | 1979 |
| M86243 | 25962 | 1982 |
| M132049 | 39751 | 1983 |
| M216091 | 65050 | 1985 |
| 391581×2216193 − 1 | 65087 | 1989 |
| M756839 | 227832 | 1992 |
| M859433 | 258716 | 1994 |
| M1257787 | 378632 | 1996 |
| M1398269 | 420921 | 1996 |
| M2976221 | 895932 | 1997 |
| M3021377 | 909526 | 1998 |
| M6972593 | 2098960 | 1999 |
| M13466917 | 4053946 | 2001 |
| M20996011 | 6320430 | 2003 |
| M24036583 | 7235733 | 2004 |
| M25964951 | 7816230 | 2005 |
| M30402457 | 9152052 | 2005 |
| M32582657 | 9808358 | 2006 |
| M43112609 2^43112609 − 1 243112609 − 1 is the largest known Mersenne prime and the largest known prime number with 12,978,189 digits. Its discovery resulted from the Great Internet Mersenne Prime Search , and won its discoverers $100,000 and a Cooperative Computing Award from the Electronic Frontier... |
12978189 | 2008 |