 Mathematics

# Largest known prime number

Print edition : March 04, 2016

ON January 7, the Great Internet Mersenne Prime Search (GIMPS) project of the University of Central Missouri discovered the new largest known prime number, 2-1, having 22,338,618 digits, on a university computer volunteered by Curtis Cooper, a professor at the university, for the project.

The new prime number, also known as M74207281, is calculated by multiplying together 74,207,281 twos and then subtracting one. It is almost 5 million digits larger than the previous record prime number, in a special class of extremely rare prime numbers known as Mersenne primes. It is only the 49th known Mersenne prime ever discovered, each increasingly difficult to find.

The primality proof of the new prime took 31 days of non-stop computing on a PC with an Intel I7-4790 CPU. To prove that there were no errors in the prime discovery process, the new prime was independently verified using both different software and hardware.

Mersenne primes have been central to number theory since they were first discussed by Euclid about 350 BCE. Mersenne primes were named after the French monk Marin Mersenne (1588-1648), who studied these numbers more than 350 years ago. A Mersenne prime is a prime number of the form 2-1.

Mersenne made a famous conjecture that the numbers 2-1 were prime for N = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and were composite for all other positive integers n≤257. It was eventually shown, after three centuries and the availability of new techniques, that Mersenne’s conjecture contained five errors, namely two are composite (N = 67, 257) and three omitted primes (N = 61, 89, 107). The correct list is: n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107 and 127.

Euclid proved that every Mersenne prime generates a “perfect number”. A perfect number is one whose proper divisors add up to the number itself. The smallest perfect number is 6 = 1 + 2 + 3 and the second perfect number is 28 = 1 + 2 + 4 + 7 + 14. Euler (1707-1783) proved that all even perfect numbers come from Mersenne primes. The newly discovered perfect number is 2 x (2-1). This number is over 44 million digits long! It is still unknown if any odd perfect numbers exist.

GIMPS, which was founded in 1996 by George Woltman to discover new world record size Mersenne primes, has discovered all 15 of the largest known Mersenne primes.

R. Ramachandran