As of January 2018 the world has a new largest prime number, affectionately known as M77232917.
It was achieved by GIMPS, Great Internet Mersenne Prime Search, a distributed computer program based on the Mersenne primes numbers formula. That formula for deriving large prime numbers is: one less than a power of two. Mersenne is where the M comes from in M77232917. The 77232917 is the power to which two is raised. In other words, this prime is produced by multiplying 77,232,917 twos together and then subtracting one. To see a picture of the exponential form written on a blackboard in chalk, see here.
This latest largest prime number has 23,249,425 digits.
Listen to Adam Spencer give an entertaining TED talk on “Why I fell in love with monster prime numbers.”