There are infinitely many prime numbers.
Assessment
The claim traces to reliable primary sources through a clear chain of evidence.
The infinitude of the primes is Euclid's theorem, proved in Book IX of the Elements around 300 BC and among the oldest and most thoroughly re-examined results in mathematics. The classical proof observes that for any finite list of primes, one more than their product has a prime divisor, and that divisor is on none of the list; it rests only on the fact that every integer greater than 1 has a prime divisor.
Several independent proofs by entirely different methods confirm the result. Euler's analytic proof shows that the sum of the reciprocals of the primes diverges, which a finite set of primes could not produce; Furstenberg gave a topological proof, Erdős a counting proof, and there are proofs from Fermat numbers and from the irrationality of pi. The theorem is formalized in the Lean Mathlib library, where the statement that the set of primes is infinite is a library theorem. No informed party disputes the result, and the stronger question of how the primes are distributed, not whether they run out, is where the live mathematics lies.
Full reasoning: the evidence and decisions behind this verdict
The claim is a theorem with an accepted proof that has stood for over two millennia without objection, satisfying the "accepted proof" route to a verified status. Euclid's argument depends on one lemma, that every integer greater than 1 has a prime divisor, which follows by taking the least divisor greater than 1 (any proper divisor of that divisor would be a smaller divisor of the original number). Granting the lemma, the inference is valid: for primes p1, ..., pk, the number N = p1 ... pk + 1 leaves remainder 1 on division by each pi, so its prime divisor is a new prime. The argument does not claim N itself is prime, a common misreading that does not affect validity.
Euler's route corroborates independently: the sum of the reciprocals of the primes diverges is itself a proven theorem (Euler 1737, with elementary proofs by Erdős and others and Mertens's quantitative refinement), and a finite sum of positive reals cannot diverge.
Formal evidence: a formal statement of the claim, that the set of natural numbers satisfying Mathlib's primality predicate is infinite, was drafted and elaborated against Mathlib v4.33.0 and recorded for second-pass review. A one-line proof from Mathlib's own library theorem (Nat.infinite_setOfPred_prime) was submitted to the checker; it compiled with zero errors and an empty axiom list, and was rejected only because the proof declaration did not carry the checker's required target name. The run's check budget was exhausted before a resubmission under the correct name, so no accepted check row exists yet; the compile-gate pass nonetheless confirms mechanically that Mathlib proves exactly the recorded statement. A later pass can complete the accepted check cheaply.
The single recorded instance, a 2013 Quanta Magazine feature by Erica Klarreich, affirms the claim as background and attributes it to Euclid without giving the proof; it adds nothing to, and takes nothing from, a verdict that rests on the proof itself. Nothing would change the conclusion short of an inconsistency in elementary arithmetic.
Decomposition
How this claim breaks down: each argument is stated as it runs, with its subclaims linked inline. ↗︎ opens a subclaim; the map shows how they fit together.
Take any finite list of primes and let N be one more than their product. Because every integer greater than 1 has a prime divisor, N has a prime divisor, and that prime cannot be on the list, since each listed prime leaves remainder 1 when dividing N; therefore no finite list contains every prime.
The inference is valid: one more than the product of any finite list of primes leaves remainder 1 on division by each of them, so any prime divisor it has is new. The argument's only premise is that every integer greater than 1 has a prime divisor, a textbook lemma no one disputes, so the argument establishes the claim on its own.
If there were only finitely many primes, the sum of their reciprocals would be a finite sum and hence finite; because the sum of the reciprocals of the primes diverges, the primes cannot be finite in number.
The inference is valid: a finite set of primes would give a finite sum of reciprocals, so divergence rules out finiteness. The argument stands or falls with the divergence of the sum of prime reciprocals, Euler's theorem, which is itself settled with several independent proofs; the argument is an independent corroboration of Euclid's rather than a dependency of it.
Provenance
Where this claim has been said, linked to its canonical form.
Euclid, who proved more than 2,000 years ago that there are infinitely many of them
On prime numbers being the atoms of arithmetic.
Asserted without evidence of the source's own. A science journalism feature that states Euclid's theorem in passing as settled background, attributing it to Euclid without presenting the proof. Ordinary for a news source; the claim's standing rests on the proof itself, not on this article.
Cite this claim: a formal citation with its evidence attached
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Created by extractor · Sep 14, 2026. Every judgment on this page is accompanied by a reasoning trace.