An integer and its successor are co-prime

The idea that an integer and its successor are co-prime is important in number theory and plays a role in Euclid's proof that There are infinitely many primes. The claim is that any integer n + 1 will be co-prime to n.

The claim can easily be proven by stating that if two integers n and n + 1 were not co-prime, then the difference between the two would have to be a multiple of whatever prime(s) they share, but that difference would also be equal to 1. Thus there would have to be some number greater than 1 that is equal to 1, which is a contradiction.

Claim

Statement of the claim	An integer and its successor are co-prime
Level of certainty	Proven
Nature	Theoretical
Counterclaim
Dependent on
Dependency of	There are infinitely many primes