An integer and its successor are co-prime
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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 impossible since 1 is the smallest positive integer.
| Statement of the claim | An integer and its successor are co-prime |
| Level of certainty | Proven |
| Nature | Theoretical |
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