# Kummer's Restatement of Euclid's Proof

### By Chris Caldwell

Euclid may have been the first to give a proof that there are infintely many primes. Even after 2000 years it stands as an excellent model of reasoning. Kummer gave a more elegant version of this proof which we give below (following Ribenboim [Ribenboim95, p. 4]). See the page "There are Infinitely Many Primes" for several other proofs.

**Theorem.**- There are infinitely many primes.
**Proof.**- Suppose that there exist only finitely many primes
**p**_{1}<**p**_{2}< ... <**p**. Let_{r}**N**=**p**_{1}^{.}**p**_{2}^{.}...^{.}**p**. The integer_{r}**N**-1, being a product of primes, has a prime divisor**p**in common with_{i}**N**; so,**p**divides_{i}**N**- (**N**-1) =1, which is absurd! ∎

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