Existing in Mathlib · Number theory

Bertrand's Postulate

For every positive natural number n, at least one prime lies strictly above n and no higher than twice n.

Bertrand's postulate
prime gaps
central binomial coefficient
prime factorization
explicit finite verification

Exact theorem

Exact Mathlib statement

theorem Nat.exists_prime_lt_and_le_two_mul (n : ℕ) (hn0 : n ≠ 0) : ∃ p : ℕ, Nat.Prime p ∧ n < p ∧ p ≤ 2 * n

The theorem at a glance

Bertrand's Postulate — a prime between n and 2n

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory number-theory poster highlights the half-open interval from n to 2n, a prime inside it, competing bounds for the central binomial coefficient, and a separate finite-case ladder. — A central-binomial contradiction settles all sufficiently large n, and an explicit descending prime cover completes the positive cases.

Statement structure

From hypotheses to conclusion

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

On an engraved number line, an open lower gate and a closed doubled upper gate contain a faceted prime token, while a central-binomial arch rises behind the interval. — For every positive n, some prime lies strictly after n and at or before 2n.

Why it matters

A mathematical landmark

Bertrand's postulate is a classical guarantee that primes occur in every interval from a positive integer to its double. Mathlib retains the distinctive Erdős proof through central binomial coefficients and an explicit finite tail.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's positive-natural form: for n ≠ 0, there exists a prime p with n < p and p ≤ 2n. The lower endpoint is strict and the upper endpoint is inclusive. It does not assert uniqueness, a prime-counting estimate, or a smaller universal interval.

The exact interval is n < p ≤ 2n; the upper endpoint is not strict.
The theorem requires n ≠ 0.
It asserts existence of at least one prime, not uniqueness or an exact count.
It does not claim an optimal prime-gap bound.
ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics.

Source and local evidence

Where the theorem comes from

Existing declaration: Nat.exists_prime_lt_and_le_two_mul in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.bertrand-postulate.v001
Source: Open the pinned upstream reference