Distributed Computing is Searching for the Largest Prime Number

The largest prime number ever discovered is (2^57,885,161)-1. It was discovered on January 25, 2013, as of the date of the writing of this article (May 6, 2013). It won the $100,000 prize awarded by the Electronic Frontier Foundation. It was found by GIMPs (great Internet prime search). It has 17,425,175 digits. The previous record prime of 12,978,189 digits was found on August 23, 2008.

They have found the top ten largest prime numbers ever found since November 14, 2001, when they found a 4,053,946 digit prime. Numbers of the form (2^n)-1, called Mersenne numbers, have dominated the search for large primes. The exponent n must be a prime number for the Mersenne number to be prime. The first Mersenne primes are (2^2)-1=3, (2^3)-1=7 and (2^5)-1=31.

The primorial is the product of consecutive prime numbers starting at 2. The primorial for 3 is, for example, 2×3=6. The primorial for 7 is 2x3x5x7=210. The primorial is useful because it eliminates all the prime numbers in the primorial if any integer greater than the largest integer in the primorial is added. For example, adding 11 to the 7 primorial results in 210+11=221. Although not a prime number because it is equal to 13 times 17, 221 is not divisible by 2, 3, 5, or 7. The next odd integer, 13, when added to 210, generates the prime number 223. The largest primorial prime ever found was found on February 28, 2012. It is 476,311 digits long and was found by James P. Burt of the Cayman Islands. The prime number is 1098133#-1, where # means the primorial ending in the prime number 1098133.

Another form of interest, but not as successful as the others, is Fermat primes. They are a power of two plus one instead of a power of two minus one like the Mersenne primes. For a Fermat prime to be prime, the exponent must be a power of two. The first five Fermat primes using n=power of two are prime:  F0=(2^2^0)+1=3,  F1=(2^2^1)+1=5, F2=(2^2^2)+1=17, F3=(2^2^3)+1=257 and f4=(2^2^4)=65,537. No other Fermat primes have been found.

The Egyptian mathematician Euclid proved in his book “The Elements” that Mersenne primes form all even perfect numbers. A perfect number is any number whose divisors add up to itself. The first perfect number is 6 because its divisors 1, 2, 3 equal 6 when added together. 

The generalized Fermat prime search has been very successful. The form is an integer raised to the power of 2^n, then one added:  k^(2^n)+1. The twelfth largest prime number ever found is a Generalized Fermat prime:  475856^(2^19)+1. It is 2,976,633 digits.