
New record prime: 2^{77,232,917}1 with 23,249,425 digits by Pace, Woltman, Kurowski, Blosser & GIMPS (26 Dec 2017). 

Perhaps the most rediscovered result about primes numbers is the following: I found that every prime number over 3 lies next to a number divisible by six. Using Matlab with the help of a friend, we wrote a program to test this theory and found that at least within the first 1,000,000 primes this holds true. Checking a million primes is certainly energetic, but it is not necessary (and just looking at examples can be misleading in mathematics). Here is how to prove your observation: take any integer n greater than 3, and divide it by 6. That is, write n = 6q + r where q is a nonnegative integer and the remainder r is one of 0, 1, 2, 3, 4, or 5.
So if n is prime, then the remainder r is either
Remember that being one more or less than a multiple of six does not
make a number prime. We have only shown that all primes other than
2 and 3 (which divide 6) have this form. 
Another prime page by Chris K. Caldwell <caldwell@utm.edu> 