Around 300 BC, Euclid proved that there are infinitely many prime numbers. Let 2, 3, ... and p be all prime numbers not greater than p, and q = 2 * 3 * ... * p+1. It is easy to see that q is not a multiple of 2, 3, …, p, because the smallest positive divisor of q must be a prime number, so either q itself is a prime number or q can be divisible by some two prime numbers between p and q [for example, 2 * 3 * 5 * 7 *1*13]. So there must be a prime number greater than p, which means there are infinitely many prime numbers.

Prime number plays an extremely important role in natural numbers, but its change is very irregular. So far, people have not found, and probably can't find a useful formula that can represent all prime numbers. The original research method is to discover the properties of prime number distribution by observing the prime number table. The existing list of perfect prime numbers was compiled by D.B. Zagale in 1977, which lists all prime numbers not exceeding 50000000. As can be seen from the prime number table, there are 25 prime numbers between 1 and 100, 1 68 prime numbers between 1000 and 135 between 1000 and 2000. There are 120 prime numbers between 3000 and 4000,19 prime numbers between 4000 and 5000, and 560 prime numbers between 5000 and 10000. It can be seen that the distribution of prime numbers is rarer as it goes up.