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Proof insight · source-linked explanation

The extra one breaks every small prime divisor

A reusable part of the proof of prime infinitude, grounded in the cited note and mathlib source.

Independent review pending

The proof idea

The key move is not to find the next prime directly. Construct n! + 1, take one of its prime factors, and notice that any prime at most n would divide both n! and n! + 1. That impossible overlap supplies a prime beyond the alleged bound.

This explanation is a separately authored interpretation. A checked Lean declaration does not review its mathematical wording.

Cited material

Theorem wording and proof ideaPrime numbers do not run out · lines 9–15Nat.exists_infinite_primes and factorial proofMathlib: infinitude of primes · lines 31–42

Use this idea

The follow-up task shifts the input to the witness theorem by one to obtain a strict bound.

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