Primes are interesting numbers. They are fundamentally simple, and yet it is difficult to anticipate them.
A prime is a number that has no divisor other than itself and 1.
- 2 = 2 x 1 = Prime
- 3 = 3 x 1 = Prime
- 4 = 2 x 2 = Not prime (and likewise for all other even numbers)
- 5 = 5 x 1 = Prime
- 7 = 7 x 1 = Prime
- 9 = 3 x 3 = Not prime
- 11 = 11 x 1 = Prime
- 13 = 13 x 1 = Prime
- 15 = 3 x 5 = Not prime
So it appears that there are patterns: initially every number appears to be prime, until you get to 4; then odd numbers, until you get to 9. But the further you go, the more you look, the more elusive they appear to be.
Maybe n!+1 is prime? But then 25 (2 x 3 x 4 + 1) is not.
What about a function P(n) = 1 + 2 x 3 x 5 x 7 x … x (the nth prime)? Well, yes, but you’re missing a bunch of primes between P(10) = 6,469,693,231, and P(11) = 200,560,690,131. Well, it works in that it gives primes, but it doesn’t give all primes. But half right is better than all wrong.
The Sieve of Eratosthenes, which is explained in this Wiki article, does a great job but is very labor intensive, and doesn’t really “predict” primes, only identifies them. No forecasting ability; no thorough formulaic process.
So, this article about the number spiral approach is interesting, in that it uses graphics to help us identify patterns. It again is not a thorough process, but it does help identify a group of primes.
Mathematicians are also looking into converting to other basis to help in identifying and predicting primes more easily. More visualizing the binary representation (converting the prime number to base 2) in this article.
And, of course, the Wiki article on prime numbers provides a wealth of information and links to several branches of current research.