Primes are fundamental parts of natural numbers.

A prime is a natural number greater than 1 that cannot be divided by any other number apart from 1 and itself. Every natural number greater than 1 can be uniquely expressed as a product of prime numbers. This result is known as the Fundamental Theorem of Arithmetic.

There are infinitely many prime numbers.

The following argument by contradiction shows the infinitude of primes:

Suppose there are finitely many primes. Let p be the product of all primes. Thus, p is divisible by all primes. The number p + 1 is not divisible by any divisor of p. Therefore, p + 1 is prime and does not divide p, which is a contradiction. Hence, the hypothesis that there are a finite number of primes was wrong. So, it can be stated that there are infinitely many primes.

Distribution of Primes

The distribution of primes among the integers appears to be random, but it follows an asymptotic pattern.

The Prime Number Theorem

It states that the number of primes less than a given number N is approximately equal to the natural logarithm of N.

So, basically, this theorem says that primes become rarer as they grow larger.

Riemann Hypothesis

It posits that the Riemann zeta function has its non-trivial zeros only at the negative even integers and complex numbers with a real part of 1/2.

This hypothesis predicts tighter bounds on the error term in the Prime Number Theorem, so its validity would lead to a closed formula for finding prime numbers. However, it has not yet been proven.