Is there a finite number of prime numbers, or an infinite number? This is a very fundamental question. In fact, there are infinitely many prime numbers, but this is not an obvious fact. The first proof of the infinitude of prime numbers appeared in Euclid's famous work Elements, in Proposition 20 of the ninth book. Euclid used the method of proof by contradiction. Assuming that there are only k prime numbers, numbered from 1 to k, and considering P = p1p2...pk + 1, it is obvious that pk > pk^, so pk must be a composite number. If pk is a prime factor of pk, then p must be one of p1, p2, ..., pk, so p must divide 1. This is a contradiction. This proves that there are infinitely many prime numbers.
Another proof of the infinitude of prime numbers was given by Euler. Similarly, using the method of proof by contradiction, first assume that there are only k prime numbers, numbered from 1 (=2) to pk, where k is a positive integer. Examining the product, it is obvious that N is a finite positive number. Using the formula for the sum of a geometric series and the fundamental theorem of arithmetic, every integer greater than 1 can be written as the product of some powers of p1, p2, ..., pk. Therefore, after removing all the brackets from the first formula, we obtain However, Euler already knew that the sum of the reciprocals of the right-hand side of the above equation is infinite, which is a contradiction. Thus, it is proved that there are infinitely many prime numbers.
If there were only a finite number of prime numbers, many difficult problems in number theory would not be considered difficult (and even the Goldbach conjecture would no longer exist). However, this would greatly reduce the charm and research value of number theory. Although Euler's proof is slightly more complex than Euclid's, it inspired the German mathematician Riemann to study the distribution of prime numbers using the zeta function.
After proving that there are infinitely many prime numbers, we can ask a more profound question: Given a large number x on the number line, how many prime numbers not exceeding x are there? At the end of the 18th century, Gauss and Legendre conjectured that the number of prime numbers not exceeding x is approximately x/lnx, and the closer x is to infinity, the more accurate this approximation becomes. Gauss's conjecture gave the rate of increase in the number of prime numbers not exceeding x as x increases.
To prove this conjecture, Riemann proposed and developed the theory of the zeta function. Riemann's work was clearly inspired by Euclid's proof of the infinitude of prime numbers. Following Riemann's direction, more than 100 years after Gauss's conjecture was proposed, in 1896, the French mathematician Hadamard and the Belgian mathematician de la Vallée Poussin each gave proofs, and since then, this conjecture has been called the Prime Number Theorem.
In the process of developing the theory of the zeta function, Riemann put forward the famous Riemann Hypothesis, which has not been proven to this day. The Riemann Hypothesis is closely related to the distribution of prime numbers, and one of its equivalent forms is that the Prime Number Theorem has the best possible error term.