How many prime numbers are there

History — How many primes are there?

Euclid proved (Elements, Book IX, Prop. 20) that the primes are infinite: if $p_1,\ldots,p_k$ were all the primes, the number $N=p_1\cdots p_k+1$ would not be divisible by any of them, so it would have a new prime factor — a contradiction.

The prime number theorem, conjectured by Gauss and proved independently by Hadamard and by de la Vallée Poussin in 1896, refines the picture: $\pi(N)\sim N/\ln N$, where $\pi(N)$ is the number of primes $\le N$. For $N=10^6$ there are $78\,498$ primes against the estimate $10^6/\ln 10^6\approx 72\,382$: a relative error of about $8\%$, which decreases (slowly) to zero as $N\to\infty$.

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Topics: Numbers and operations
Concepts: Prime number
People: De la Vallée Poussin · Euclid · Carl Friedrich Gauss · Hadamard