Proved
Proved declaration Nat.exists_infinite_primes ↗
The cited source item and version only; this is not an independent OpenMath assessment.
mathlib · infinitude of primes · Mathlib/Data/Nat/Prime/Infinite.lean lines 31–42 →
Observed Oct 1, 2026
Theorem · version 3
There are infinitely many primes
Statement
The set of natural-number primes is infinite.
Assumptions
- Natural numbers
- Ordinary primality
Scope
Natural numbers; no claim of this equivalence for arbitrary ordered sets.
Evidence
Source reports
Proved
Known theorem in authored reference note ↗
The cited source item and version only; this is not an independent OpenMath assessment.
OpenMath · prime note · notes/primes.md lines 9–15, 21–29 →
Observed Oct 1, 2026
Connections
In this problem · publication v7 · Selected context
Why are there infinitely many primes? →
Equivalent formulation · Proved
A finite set of natural numbers is bounded; a bounded set lies in a finite initial segment. Thus infinitude and unboundedness are equivalent here.
A bounded set of natural numbers is finite, so infinitude and unboundedness agree.
Relationship not independently reviewed. This concerns its use in this problem.
Formalize a prime strictly above every bound → · Related problem work; no execution evidence for this statement is asserted.
In this problem · publication v8
Why are there infinitely many primes? →
Equivalent formulation · Proved
A finite set of natural numbers is bounded; a bounded set lies in a finite initial segment. Thus infinitude and unboundedness are equivalent here.
A bounded set of natural numbers is finite, so infinitude and unboundedness agree.
Relationship not independently reviewed. This concerns its use in this problem.
Formalize a prime strictly above every bound → · Related problem work; no execution evidence for this statement is asserted.
Sources
OpenMath · prime note →
- notes/primes.md lines 9–15, 21–29 → · Original passage ↗
Recorded wording
For every natural number n, does there exist a prime p with n < p?
paraphrase · Licensed source links and exact source spans; attribution retained.
OpenMath staff · Revision a640f3adfa536da0
mathlib · infinitude of primes →
- Mathlib/Data/Nat/Prime/Infinite.lean lines 31–42 → · Original passage ↗
Recorded wording
For every natural number n, does there exist a prime p with n < p?
paraphrase · Licensed source links and exact source spans; attribution retained.
mathlib contributors · Revision 32d24245c7a12ded17325299fd41d412022cd3fe
History
Statement versions
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessment history · 2
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Known theorem in authored reference note ↗
The cited source item and version only; this is not an independent OpenMath assessment.
Observed Oct 1, 2026 · Source passage →
Proved
Proved declaration Nat.exists_infinite_primes ↗
The cited source item and version only; this is not an independent OpenMath assessment.
Observed Oct 1, 2026 · Source passage →
Record details
- Claim ID
- claim-primes-infinite
- Statement version
- 3
- SHA-256
- 1c5093b35fd7fb4f8a8d26fd5f0ebc8c4ccfbdb46e2889ad9005ea6e14fabef8