Existing in Mathlib · Number theory

Infinitely Many Primes

Choose any natural number n. There is always a prime number p with p at least n, so no finite threshold contains all primes.

prime numbers
natural numbers
unboundedness
existence

Exact theorem

Exact Mathlib statement

theorem Nat.exists_infinite_primes (n : ℕ) : ∃ p, n ≤ p ∧ Nat.Prime p

The theorem at a glance

Infinitely Many Primes at a glance

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

An ivory theorem poster shows a path crossing an arbitrary natural-number threshold n and continuing to a prime beacon p at or above it, with a three-step reading guide and an unboundedness-only scope panel. — Every threshold has a prime at or above it; the poster presents unboundedness without implying a prime-gap or counting estimate.

Statement structure

From hypotheses to conclusion

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

Scientific figure illustrating: Beyond any chosen threshold, another prime can be found. — Beyond any chosen threshold, another prime can be found.

Why it matters

A mathematical landmark

Euclid's theorem is the archetypal infinitude argument in number theory. This threshold formulation states unboundedness of the prime numbers directly.

Proof architecture

A least-prime-factor route beyond every threshold

3 curated stages

Mathlib chooses p as the least prime factor of n! + 1, proves p prime, and rules out p below the arbitrary threshold because such a p would divide both n! and n! + 1 and therefore divide one. The result is unboundedness of primes, not a next-prime or prime-gap theorem.

These source-anchored stages explain the retained Mathlib proof route. They are generated explanations, not proof evidence or an extracted Lean proof-term dependency graph.

Select the least prime factor of the factorial successor
Select the least prime factor of the factorial successor
From the factorial successor, select its least prime factor; the source proves that this selected factor is prime.
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From the factorial successor, select its least prime factor; the source proves that this selected factor is prime.
p := minFac (n! + 1) and Nat.Prime p
The source defines p to be minFac (n! + 1). Factorial positivity shows n! + 1 is not one, so minFac_prime proves that this specifically selected least factor p is prime.
A factor at or below the threshold would divide one
A factor at or below the threshold would divide one
If the selected prime factor lay at or below the threshold, it would divide both the factorial and its successor, hence divide one—impossible for a prime.
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If the selected prime factor lay at or below the threshold, it would divide both the factorial and its successor, hence divide one—impossible for a prime.
p ∣ n! and p ∣ n! + 1 imply p ∣ 1, contradicting Nat.Prime p
Assume contrariwise that p lies at or below n. Positivity of p and dvd_factorial then give p ∣ n!, while minFac_dvd gives p ∣ n! + 1; the addition divisibility lemma forces p ∣ 1, contradicting the fact that p is prime.
A prime exists at or above every threshold
A prime exists at or above every threshold
The contradiction places the selected prime at or above the arbitrary threshold, establishing that primes are unbounded.
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The contradiction places the selected prime at or above the arbitrary threshold, establishing that primes are unbounded.
∃ p, n ≤ p ∧ Nat.Prime p
The divisibility contradiction proves n ≤ p. Pairing that bound with the already established primality of p yields the witness ⟨p, np, pp⟩ for the arbitrary threshold n, which is exactly the theorem's unbounded-primes conclusion.

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 is the exact threshold-existence form in Mathlib. It proves a prime exists at or above every natural-number bound; it does not give a quantitative prime-gap estimate or a count of primes below a bound.

Proof Atlas did not originate Euclid's theorem or Mathlib's declaration.
The page indexes and locally reproduces an upstream theorem.
No quantitative distribution theorem for primes is asserted.

Source and local evidence

Where the theorem comes from

Existing declaration: Nat.exists_infinite_primes in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.infinitely-many-primes.v001
Source: Open the pinned upstream reference