|
|
| Line 1: |
Line 1: |
| In [[number theory]], '''[[Hillel Furstenberg]]'s proof of the infinitude of primes''' is a celebrated [[topology|topological]] [[Mathematical proof|proof]] that the [[integer]]s contain [[Infinite set|infinitely]] many [[prime number]]s. When examined closely, the proof is less a statement about topology than a statement about certain properties of [[arithmetic sequence]]s.<ref name=mercer/> Unlike [[Euclid's Theorem#Euclid's proof|Euclid's classical proof]], Furstenberg's proof is a [[proof by contradiction]]. The proof was published in 1955 in the [[American Mathematical Monthly]] while Furstenberg was still an [[undergraduate student]] at [[Yeshiva University]].
| | Ed is what individuals contact me and my spouse doesn't like it at all. Ohio is where her home is. The favorite pastime for him and his children is to play lacross and he'll be starting some thing else along with it. Distributing manufacturing is exactly where her primary income arrives from.<br><br>my weblog; psychics online - [http://m-card.co.kr/xe/mcard_2013_promote01/29877 have a peek at this web-site] - |
| | |
| ==Furstenberg's proof==
| |
| Define a topology on the integers '''Z''', called the [[evenly spaced integer topology]], by declaring a subset ''U'' ⊆ '''Z''' to be an [[open set]] [[if and only if]] it is either the [[empty set]], ∅, or it is a [[union (set theory)|union]] of arithmetic sequences ''S''(''a'', ''b'') (for ''a'' ≠ 0), where
| |
| | |
| :<math>S(a, b) = \{ a n + b\, |\, n \in \mathbb{Z} \} = a \mathbb{Z} + b. \, </math>
| |
| | |
| In other words, ''U'' is open if and only if every ''x'' ∈ ''U'' admits some non-zero integer ''a'' such that ''S''(''a'', ''x'') ⊆ ''U''. The [[Topological space|axioms for a topology]] are easily verified:
| |
| | |
| * By definition, ∅ is open; '''Z''' is just the sequence ''S''(1, 0), and so is open as well.
| |
| * Any union of open sets is open: for any collection of open sets ''U''<sub>''i''</sub> and ''x'' in their union ''U'', any of the numbers ''a''<sub>''i''</sub> for which ''S''(''a''<sub>''i''</sub>, ''x'') ⊆ ''U''<sub>''i''</sub> also shows that ''S''(''a''<sub>''i''</sub>, ''x'') ⊆ ''U''.
| |
| * The intersection of two (and hence finitely many) open sets is open: let ''U''<sub>1</sub> and ''U''<sub>2</sub> be open sets and let ''x'' ∈ ''U''<sub>1</sub> ∩ ''U''<sub>2</sub> (with numbers ''a''<sub>1</sub> and ''a''<sub>2</sub> establishing membership). Set ''a'' to be the [[lowest common multiple]] of ''a''<sub>1</sub> and ''a''<sub>2</sub>. Then ''S''(''a'', ''x'') ⊆ ''S''(''a''<sub>''i''</sub>, ''x'') ⊆ ''U''<sub>i</sub>.
| |
| | |
| This topology has two notable properties:
| |
| | |
| # Since any non-empty open set contains an infinite sequence, a finite set cannot be open; put another way, the [[complement (set theory)|complement]] of a finite set cannot be a [[closed set]].
| |
| # The basis sets ''S''(''a'', ''b'') are [[clopen set|both open and closed]]: they are open by definition, and we can write ''S''(''a'', ''b'') as the complement of an open set as follows:
| |
| | |
| ::<math>S(a, b) = \mathbb{Z} \setminus \bigcup_{j = 1}^{a - 1} S(a, b + j).</math>
| |
| | |
| The only integers that are not integer multiples of prime numbers are −1 and +1, i.e.
| |
| | |
| ::<math>\mathbb{Z} \setminus \{ -1, + 1 \} = \bigcup_{p \mathrm{\, prime}} S(p, 0).</math>
| |
| | |
| By the first property, the set on the left-hand side cannot be closed. On the other hand, by the second property, the sets ''S''(''p'', 0) are closed. So, if there were only finitely many prime numbers, then the set on the right-hand side would be a finite union of closed sets, and hence closed. This would be a [[contradiction]], so there must be infinitely many prime numbers.
| |
| | |
| ==Notes==
| |
| {{reflist|refs=
| |
| <ref name=mercer>
| |
| {{Cite journal
| |
| | last1 = Mercer | first1 = Idris D.
| |
| | title = On Furstenberg's Proof of the Infinitude of Primes
| |
| | journal = American Mathematical Monthly
| |
| | volume = 116
| |
| | pages = 355–356
| |
| | year = 2009
| |
| | doi = 10.4169/193009709X470218
| |
| | url = http://www.idmercer.com/monthly355-356-mercer.pdf
| |
| }}</ref>
| |
| }}
| |
| | |
| ==References==
| |
| *{{Cite document | last1=Aigner | first1=Martin|author1-link=Martin Aigner | last2=Ziegler | first2=Günter M. | author2-link=Günter M. Ziegler | title=[[Proofs from The Book]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | year=1998 | postscript=<!--None-->}}
| |
| * {{cite journal
| |
| | last = Furstenberg
| |
| | first = Harry
| |
| | authorlink = Hillel Furstenberg
| |
| | title = On the infinitude of primes
| |
| | journal = [[American Mathematical Monthly]]
| |
| | volume = 62
| |
| | year = 1955
| |
| | pages = 353
| |
| | doi = 10.2307/2307043
| |
| | jstor = 2307043
| |
| | issue = 5
| |
| | publisher = Mathematical Association of America
| |
| }} {{MR|0068566}}
| |
| | |
| ==External links==
| |
| *[http://www.everything2.com/index.pl?node_id=1460203 Furstenberg's proof that there are infinitely many prime numbers] at [[Everything2]]
| |
| *{{PlanetMath|urlname=FurstenbergsProofOfTheInfinitudeOfPrimes|title=Fürstenberg's proof of the infinitude of primes}}
| |
| | |
| {{DEFAULTSORT:Furstenberg's Proof Of The Infinitude Of Primes}}
| |
| [[Category:Article proofs]]
| |
| [[Category:General topology]]
| |
| [[Category:Prime numbers]]
| |
Ed is what individuals contact me and my spouse doesn't like it at all. Ohio is where her home is. The favorite pastime for him and his children is to play lacross and he'll be starting some thing else along with it. Distributing manufacturing is exactly where her primary income arrives from.
my weblog; psychics online - have a peek at this web-site -