Number theory · existing Mathlib theorem

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

Four separate status axes

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

These states distinguish upstream identity, local reproduction, review, and Atlas acceptance. This page is part of the public, read-only Mathlib landmark collection.

Formal orientation

Statement map

This deterministic map orients the reader from the mathematical summary to the pinned declaration and exact checked form. The HTML statement below is authoritative.

The theorem at a glance

Infinitely Many Primes at a glance

A source-bound editorial overview of the exact Mathlib formulation. The image is explanatory; the statement map, formal type, and pinned source remain authoritative.

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.

A stone path passes a gate and continues toward an open horizon, marked by isolated gold beacons among irregular composite piles. — Beyond any chosen threshold, another prime can be found.

Complementary intuition

A mathematical landmark

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

The low-text schematic gives the theorem a visual identity; it does not replace the source-bound poster or exact statement.

Authoritative formal type

Exact Mathlib statement

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

Selected evidence declaration text

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

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.

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.

Provenance

Origin and evidence stay separate

Existing declaration: Nat.exists_infinite_primes in mathlib
Relationship: The checked artifact is the upstream declaration itself
Preferred checked artifact: artifact.library.mathlib.infinitely-many-primes.v001
Source handling: Verified upstream reference; no mirrored source package