Difference between revisions of "There are infinitely many primes"
| (4 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
'''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}}. | '''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 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 | 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 | {{Claim | ||
| Line 9: | Line 9: | ||
|Counterclaim=There is a finite amount of prime numbers | |Counterclaim=There is a finite amount of prime numbers | ||
|DependentOn1=An integer and its successor are co-prime | |DependentOn1=An integer and its successor are co-prime | ||
|DependentOn2=Any integer bigger than 1 is the multiple of a prime | |DependentOn2=Any integer bigger than 1 is the multiple of a prime | ||
| | |DependencyOf1= | ||
}} | }} | ||
==References== | ==References== | ||
<references/> | <references/> | ||
[[Category:Mathematics]] | |||
Latest revision as of 20:59, 22 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 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.
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.