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
StatementThe 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 Classification
Statement Form 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
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Additional Source Prime Number Theorem (Wikipedia)History section
Source Prime Number Theorem (Wikipedia)
Proved By
Source Prime Number Theorem (Wikipedia)
Source Prime Number Theorem (Wikipedia)
Sources
1. Prime Number Theorem (Wikipedia)
Wikimedia Foundationlead 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)
history section
independently proved by Jacques Hadamard and Charles Jean de la Vallee Poussin in 1896
lead section
the prime number theorem (PNT) describes the asymptotic distribution of prime numbers among the positive integers.
View the Source Reader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.