GIMPS Discovers The Largest Known Prime Number Yet

The Great Internet Mersenne Prime Search (GIMPS) discovered the largest prime number yet on January 7, 2016. The number is 2^{74, 207, 281} - 1. It contains 22,338,618 digits. If you are wondering how long it is, suppose that you can write an average of 2 digits per second, you will be able to finish writing this number in about 129 days without eating, sleeping, or toilet break. Using the same rate, if you are going to write this number for 6 hours a day, then you will finish it in about 517 days (roughly one and a half years).

GIMPS has been calculating large prime numbers since 1996 and has discovered 15 of the largest known prime numbers. As of this writing, the 11 largest prime numbers are Mersenne primes. Mersenne primes are prime numbers of the form 2^n - 1 for some integer n. Three of the smallest Mersenne primes are 2^2 - 1 = 3, 2^3 - 1 = 7, and 2^5 - 1 = 31 .

Question: Are all positive integers of the form 2^n - 1 prime numbers?

For the non-math persons, prime numbers are positive integers that can only be divided by 1 and itself. For example, 5 is a prime number since you cannot divide 5 by any number except one and itself. On the other hand, 8 is not a prime number because 8 can be divided by 1, 2, 4, and 8.

It was already proven by Euclid (some 2300 years ago) that there is no largest prime number, so the search for large prime numbers will never end.

Some of the most useful application of prime numbers is cryptography, particularly internet security. It’s what makes your password safe. It is what makes shopping safe. Well, you still have to be careful though.

Mersenne Primes Under the Microscope

What are Mersenne Primes?

The recent discovery of the largest known prime number which is 17 million digits highlighted the importance of Mersenne primes.  This newest prime number and the thirteen largest primes are all Mersenne primes. But what are Mersenne primes really and why are they important in finding the largest prime numbers?

If a prime number can be expressed in the form 2^n - 1, n an integer, then it is said to be a Mersenne prime. The name came from Marin Mersenne who studied them, a French monk and well-known mathematician in the 17th century.


Many early mathematicians saw and conjectured that if n is a prime number, 2^n - 1 is prime.  For instance, in the figure above, for n = 2, 3, 5, 7 which are prime numbers, their corresponding 2^n - 1 values are also prime numbers. However, this is not true in general since 2^{11} - 1 = 2047 = (23)(89) is not prime.  » Read more