Difference between revisions of "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|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 idea that '''an integer and its successor are co-prime''' is important in [[Number theory|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 | 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 | {{Claim | ||
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|Counterclaim= | |Counterclaim= | ||
|DependentOn1= | |DependentOn1= | ||
| | |DependencyOf1=There are infinitely many primes | ||
}} | }} | ||
[[Category:Mathematics]] | |||
Latest revision as of 20:59, 22 January 2022
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.
| Statement of the claim | An integer and its successor are co-prime |
| Level of certainty | Proven |
| Nature | Theoretical |
| Counterclaim | |
| Dependent on |
|
| Dependency of |