I was wondering if there was some way to determine if a number is prime without testing every number below it to see if it has a divisor.

Such as, I found that all primes can be expressed in the form: $\displaystyle 6k\pm1$ (except for 2 and 3) but not all numbers that can be expressed in $\displaystyle 6k\pm1$ are prime.

Is there something that shows definitively that a number is prime?