Difference between revisions of "There are infinitely many primes"
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'''There are infinitely many primes''' is a claim of number | '''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.{{Cite web|url=https://primes.utm.edu/notes/proofs/infinite/euclids.html|title=Euclid's Proof of the Infinitude of Primes (c. 300 BC)|last=Caldwell|first=Chris K.|date=Unknown Publication Date|publisher=Prime Pages, University of Tennessee at Martin|access-date=January 17, 2022}}. | ||
The proof consists in imagining the series of all prime numbers if that series was finite. Multiplying all those prime numbers together and subtracting 1 from the product, one would necessarily arrive at a very large number which would also be co-prime to all other prime numbers. Thus that number would have to be a multiple of some other primes, thus showing that the initial series of prime number was incomplete. | The proof consists in imagining the series of all prime numbers if that series was finite. Multiplying all those prime numbers together and subtracting 1 from the product, one would necessarily arrive at a very large number which would also be co-prime to all other prime numbers. Thus that number would have to be a multiple of some other primes, thus showing that the initial series of prime number was incomplete. Iterating this operation, this means that there can be no finite set of all prime numbers. | ||
{{Claim | {{Claim | ||
Revision as of 03:14, 18 January 2022
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 in imagining the series of all prime numbers if that series was finite. Multiplying all those prime numbers together and subtracting 1 from the product, one would necessarily arrive at a very large number which would also be co-prime to all other prime numbers. Thus that number would have to be a multiple of some other primes, thus showing that the initial series of prime number was incomplete. Iterating this operation, this means that there can be no finite set of all prime numbers.
| Statement of the claim | There are infinitely many primes |
| Level of certainty | Proven |
| Nature | Theoretical |
| Counterclaim | There are a finite amount of prime numbers |
| Dependent on |
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| Dependency of |
References
- ↑ 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.