Euclid's proof
The argument
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.
Does it work?
Not yet evaluated.
Premises
- UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →Every integer greater than 1 has a prime divisor.
Where it is made
Passages where sources make, criticize or mention this argument. They record where it appears, not how strong it is, and none has precedence over another.
No source is recorded as making it.
Earlier ratings
- Sep 16, 2026
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.