FAQ: Are all primes (past 2 and 3) of the forms 6n+1 and 6n-1?

By Chris Caldwell

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.

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