There are infinitely many primes

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..

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.

Claim

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