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Theorem

Prime Number Theorem

Also Known As PNT
Number Theory
Step Count Approaching a Smooth Curve

The prime number theorem describes how prime numbers are distributed among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/

Facts
Statement
The number of primes less than or equal to x, written pi(x), grows asymptotically like x divided by the natural logarithm of x, so their ratio approaches 1 as x grows large; equivalently, the average gap between consecutive primes near N is roughly the natural logarithm of N. 1
Proof Year
1896 1
Classification
Statement Form
Inequality 1
Connections

Associated With

Prime Number, Concepts

The theorem quantifies how the density of prime numbers thins out among the integers.

Additional Source Prime Number Theorem (Wikipedia)lead paragraph

Has Statement Form

Inequality, Concepts

In Branch

Analytic Number Theory, Branches of Mathematics
Additional Source Prime Number Theorem (Wikipedia)History of the proof of the asymptotic law of prime numbers
Number Theory, Branches of Mathematics
Source Prime Number Theorem (Wikipedia)

Proved By

Charles-Jean de la Vallee Poussin, Mathematicians
Source Prime Number Theorem (Wikipedia)
Jacques Hadamard, Mathematicians
Source Prime Number Theorem (Wikipedia)
Sources
1. Prime Number Theorem (Wikipedia)
Wikimedia Foundation
  • lead paragraph
    It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs.
  • body text
    the average gap between consecutive prime numbers among the first N integers is roughly log(N)
  • Lead section
    The theorem was proved independently by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function).
  • lead section
    the prime number theorem (PNT) describes the asymptotic distribution of prime numbers among the positive integers.
  • In Branch: Analytic Number Theory, History of the proof of the asymptotic law of prime numbers
    In particular, it is in this paper that the idea to apply methods of complex analysis to the study of the real function π(x) originates.
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