Prove or disprove Every prime number except 2 and 3 has the

Prove or disprove: Every prime number except 2 and 3 has the form 6q + 1 or 6q + 5.

Solution

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.

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.