1. The Largest Known Prime Number
Kun-Mao Chao (趙坤茂)
Department of Computer Science and Information Engineering
National Taiwan University, Taiwan
A note given in BCC class on May 28, 2014

2. Mersenne prime a Mersenne prime is a prime number of the form Mn=2n-1.
Since 1997, all newly found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search” (GIMPS), a distributed computing project on the Internet.

3. The ten largest known prime numbers (as of May 28, 2014)
Lucas–Lehmer primality test

4. AKS primality test
A deterministic primality-proving algorithm created and published by Manindra Agrawal, Neeraj Kayal, and Nitin Saxena, computer scientists at the Indian Institute of Technology Kanpur, on August 6, 2002.
The algorithm determines whether a number is prime or composite within polynomial time. "PRIMES is in P."