Largest known prime

The largest known prime as of February 2013 is \(2^{57,885,161} − 1\). A prime number is an integer greater than 1 that has no divisors other than 1 and itself. It is well known that there are infinitely many prime numbers (as proven by Euclid), so the search for very large s is limitless. The gives monetary prizes to people who discover new large primes.

Records
The fastest known algorithm for finding large prime numbers is the, which tests Mersenne primes. Thus the largest known primes have been Mersenne primes for a long time. George Woltman's distributed computing program GIMPS, an implementation of the Lucas-Lehmer test, has found all the new records since 1996.

A list of record primes (as of January 6, 2015) is given below:

Proof of the infinitude of primes
Euclid gives an elegant proof that there are infinite prime numbers.

Suppose there is a finite number of prime numbers ..., and let their product be P. Then P + 1 is one more than a multiple of, and one more than a multiple of , etc. P + 1 is not divisible by any of our primes, and thus it has no prime factors. Since P + 1 > 1, this is impossible.