There are infinitely many primes

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There are infinitely many primes is a claim in number theory which states that any finite set of prime numbers will be followed by some larger prime number somewhere along the number line. The claim was proven, among other ways, through Reductio ad absurdum. The first known instance of a proof dates back to Euclid, c. 300 B.C.[1].

The proof consists of assuming the series of all prime numbers to be finite. Multiplying all those prime numbers together and adding 1, one would arrive at an integer bigger than 1 which would be co-prime to all prime numbers. Thus that number would have to be a multiple of some other prime(s), showing that the initial series of prime numbers was incomplete. This is a contradiction, so the initial assumption has to be false.

Statement of the claim There are infinitely many primes
Level of certainty Proven
Nature Theoretical
Counterclaim There is a finite amount of prime numbers
Dependent on

An integer and its successor are co-prime
Any integer bigger than 1 is the multiple of a prime

Dependency of


  1. Caldwell, Chris K. (Unknown Publication Date) Euclid's Proof of the Infinitude of Primes (c. 300 BC). Prime Pages, University of Tennessee at Martin. Accessed on January 17, 2022.