Is there a pattern for prime numbers? - Answers

So far, the best and most general pattern found is that, over three, all prime numbers are of the form 6n +/- 1. In other words, they're either 6n - 1 or 6n + 1, for some n. Here is why this is true. We could do a proof by contradiction and assume that all the natural numbers greater than or equal to 5 are prime. (of course they are not!) We start with5 which is 6-1. The numbers would then be 6n - 1, 6n, 6n + 1, 6n + 2, 6n + 3, 6n + 4, and 6n + 5 for some natural number n. If it is 6n, then the number is divisible by 6. When it is 6n + 2, the number is the same as 2(3n+1) so it is divisible by 2. Consider 6n + 3, the number is 3(2n+1), so it is divisible by 3. Last look at 6n + 4, the number is divisible by 2, for it's 2(3n + 2). Therefore all numbers of the form 6n, 6n + 2, 6n + 3, and 6n + 4 are not prime. The only possibilities this leaves are 6n - 1 and 6n + 1. This entire thing can be written more elegantly with congruences, but the goal here was simplicity! There are many other patterns in primes. See the attached link to see them.