Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?  (from the Prime Pages' list of frequently asked questions)
 New record prime: 277,232,917-1 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 non-negative integer and the remainder r is one of 0, 1, 2, 3, 4, or 5. If the remainder is 0, 2 or 4, then the number n is divisible by 2, and can not be prime. If the remainder is 3, then the number n is divisible by 3, and can not be prime. So if n is prime, then the remainder r is either 1   (and   n = 6q + 1   is one more than a multiple of six), or 5   (and   n = 6q + 5 = 6(q+1) - 1   is one less than a multiple of six). 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