Largest known prime

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The largest known prime is the largest integer that is currently known to be a prime number.

Graph of number of digits in largest known prime by year, since the electronic computer. Note that the vertical scale is logarithmic.

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 enjoy searching for large prime numbers. This may also be profitable as there are several prizes offered by the Electronic Frontier Foundation for record primes.^[1]

Because the FFT implementation of the Lucas–Lehmer test for Mersenne numbers is faster than other primality tests for other kinds of primes, many of the largest known primes are Mersenne primes. As of December 2007 there were six Mersenne primes among the ten largest known primes.^[2] The last 13 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.^[3]

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.^[4]

As of March 2008, the largest known prime was discovered by the distributed computing project Great Internet Mersenne Prime Search (GIMPS):

2^32,582,657 − 1.

This was confirmed to be a prime number on September 11, 2006. This number is 9,808,358 digits long and is the 44th known Mersenne prime.

Its predecessor as largest known prime, 2^30,402,457 − 1, was first shown to be prime on December 15, 2005 by GIMPS also. GIMPS found the 10 latest records on ordinary computers operated by participants around the world.

There is a $100,000 prize for the first known prime with at least 10,000,000 digits. A Mersenne prime 2^p − 1 with p ≥ 33,219,281 would have at least 10,000,000 digits, and GIMPS is testing many candidates of this size.