How can we prove that a number is a prime with some limits?

Question

A simple python program implemented a simple fact that the number is prime if no numbers within it's square root including the square root and excluding 1 divide it. Empirically analyzing it for some 50 numbers I found it true and the remaining numbers that were not divided by any numbers as mentioned were prime numbers. But such observation didn't fully satisfy me? How can we prove this using more properties and theories that have already been verified(in an abstract way that is)?