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| In [[mathematics]], the '''Kummer–Vandiver conjecture''', or '''Vandiver conjecture''', states that a prime ''p'' does not divide the [[class number (number theory)|class number]] ''h<sub>K</sub>'' of the maximal real [[subfield]] <math>K=\mathbb{Q}(\zeta_p)^+</math> of the ''p''-th [[cyclotomic field]].
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| The conjecture was first made by [[Ernst Kummer]] in 1849 December 28 and 1853 April 24 in letters to [[Leopold Kronecker]], reprinted in {{harv|Kummer|1975|loc=pages 84, 93, 123–124}}, and independently rediscovered around 1920 by [[Philipp Furtwängler]] and {{harvs|txt|authorlink=Harry Vandiver|first=Harry |last=Vandiver|year=1946|loc=p. 576}},
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| As of 2011, there is no particularly strong evidence either for or against the conjecture and it is unclear whether it is true or false, though it is likely that counterexamples are very rare.
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| ==Background==
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| The class number ''h'' of the cyclotomic field <math>\mathbb{Q}(\zeta_p)</math> is a product of two integers ''h''<sub>1</sub> and ''h''<sub>2</sub>, called the first and second factors of the class number, where ''h''<sub>2</sub> is the class number of the maximal real [[subfield]] <math>K=\mathbb{Q}(\zeta_p)^+</math> of the ''p''-th [[cyclotomic field]]. The first factor ''h''<sub>1</sub> is well understood and can be written explicitly in terms of Bernoulli numbers, and is usually rather large. The second factor ''h''<sub>2</sub> is not well understood and seems hard to compute explicitly.
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| Kummer showed that if a prime ''p'' does not divide the class number ''h'', then Fermat's last theorem holds for exponent ''p''.
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| Kummer also showed that if ''p'' divides the second factor, then it also divides the first factor. In particular the Kummer–Vandiver conjecture holds for regular primes.
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| ==Evidence for and against the Kummer–Vandiver conjecture==
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| Kummer verified the Kummer–Vandiver conjecture for ''p'' less than 200, and Vandiver extended this to ''p'' less than 600.
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| {{harvs|txt | last1=Buhler | first1=Joe | last2=Crandall | first2=Richard | author2-link=Richard Crandall | last3=Ernvall | first3=Reijo | last4=Metsänkylä | first4=Tauno | last5=Shokrollahi | first5=M. Amin | editor1-last=Bosma | editor1-first=Wieb | title=Computational algebra and number theory. Proceedings of the 2nd International Magma Conference held at Marquette University, Milwaukee, WI, May 12–16, 1996. | doi=10.1006/jsco.1999.1011 | mr=1806208 | year=2001 | journal=Journal of Symbolic Computation | issn=0747-7171 | volume=31 | issue=1 | chapter=Irregular primes and cyclotomic invariants to 12 million | doi=10.1006/jsco.1999.1011 | pages=89–96}} verified it for ''p'' < 12 million. {{harvtxt|Harvey|2008}} extended this to primes less than 163 million.
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| {{harvtxt|Washington|1996|p=158}} describes an informal probability argument, based on rather dubious assumptions about the equidistribution of class numbers mod ''p'', suggesting that the number of primes less than ''x'' that are exceptions to the Kummer–Vandiver conjecture might grow like (1/2)log log ''x''. This grows extremely slowly, and suggests that the computer calculations do not provide much evidence for Vandiver's conjecture: for example, the probability argument (combined with the calculations for small primes) suggests that one should only expect about 1 counterexample in the first 10<sup>100</sup> primes, suggesting that it is unlikely any counterexample will be found by further brute force searches even if there are an infinite number of exceptions.
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| {{harvtxt|Mihăilescu|2010}} gave a refined version of Washington's heuristic argument, suggesting that the Kummer–Vandiver conjecture is probably true.
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| {{harvtxt|Schoof|2003}} gave conjectural calculations of the class numbers of real cyclotomic fields for primes up to 10000, which strongly suggest that the class numbers are not randomly distributed mod ''p''. They tend to be quite small and are often just 1. For example, assuming the [[generalized Riemann hypothesis]], the class number of the real cyclotomic field for the prime ''p'' is 1 for ''p''<163, and divisible by 4 for ''p''=163.
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| ==Consequences of the Kummer–Vandiver conjecture==
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| {{harvtxt|Kurihara|1992}} showed that the conjecture is equivalent to a statement in the [[algebraic K-theory]] of the integers, namely that ''K''<sub>''n''</sub>('''Z''') = 0 whenever ''n'' is a multiple of 4. In fact from the Kummer–Vandiver conjecture and the [[norm residue isomorphism theorem]] follow a full conjectural calculation of the ''K''-groups for all values of ''n''; see [[Quillen–Lichtenbaum conjecture]] for details.
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| ==See also==
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| * [[regular prime|regular and irregular primes]]
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| == References ==
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| <references/>
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| *{{Citation | last1=Buhler | first1=Joe | last2=Crandall | first2=Richard | author2-link=Richard Crandall | last3=Ernvall | first3=Reijo | last4=Metsänkylä | first4=Tauno | last5=Shokrollahi | first5=M. Amin | editor1-last=Bosma | editor1-first=Wieb | title=Computational algebra and number theory. Proceedings of the 2nd International Magma Conference held at Marquette University, Milwaukee, WI, May 12–16, 1996. | mr=1806208 | year=2001 | journal=Journal of Symbolic Computation | issn=0747-7171 | volume=31 | issue=1 | chapter=Irregular primes and cyclotomic invariants to 12 million |doi=10.1006/jsco.1999.1011 | pages=89–96}}
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| *{{Citation | last1=Ghate | first1=Eknath | editor1-last=Adhikari | editor1-first=S. D. | editor2-last=Katre | editor2-first=S. A. | editor3-last=Thakur | editor3-first=Dinesh | title=Cyclotomic fields and related topics | url=http://www.math.tifr.res.in/~eghate/vandiver.pdf | publisher=Bhaskaracharya Pratishthana, Pune | series=Proceedings of the Summer School on Cyclotomic Fields held in Pune, June 7–30, 1999 | mr=1802389 | year=2000 | chapter=Vandiver's conjecture via K-theory | pages=285–298}}
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| *{{Citation | last1=Kummer | first1=Ernst Eduard | editor1-last=Weil | editor1-first=André | editor1-link=André Weil | title=Collected papers. Volume 1: Contributions to Number Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-06835-0 | mr=0465760 | year=1975}}
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| *{{Citation | last1=Kurihara | first1=Masato | title=Some remarks on conjectures about cyclotomic fields and K-groups of Z | url=http://www.numdam.org/item?id=CM_1992__81_2_223_0 | mr=1145807 | year=1992 | journal=Compositio Mathematica | issn=0010-437X | volume=81 | issue=2 | pages=223–236}}
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| *{{Citation | last1=Mihăilescu | first1=Preda | title=Turning Washington's heuristics in favor of Vandiver's conjecture | year=2010 | arxiv=1011.6283}}
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| *{{Citation | last1=Schoof | first1=René | title=Class numbers of real cyclotomic fields of prime conductor | doi=10.1090/S0025-5718-02-01432-1 | mr=1954975 | year=2003 | journal=[[Mathematics of Computation]] | issn=0025-5718 | volume=72 | issue=242 | pages=913–937}}
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| *{{Citation | last1=Vandiver | first1=H. S. | title=Fermat's last theorem. Its history and the nature of the known results concerning it | jstor=2305236 | mr=0018660 | year=1946 | journal=[[American Mathematical Monthly|The American Mathematical Monthly]] | issn=0002-9890 | volume=53 | pages=555–578}}
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| * {{cite book | first=Lawrence C.|last= Washington | title=Introduction to Cyclotomic Fields | publisher=Springer | year=1996 | isbn=0-387-94762-0}}
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| ==External links==
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| *{{citation|first=David|last=Harvey|url=http://web.maths.unsw.edu.au/~davidharvey/papers/irregular/|title=Irregular primes to 163 million|year=2011|series=80}}
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| {{DEFAULTSORT:Kummer-Vandiver conjecture}}
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| [[Category:Cyclotomic fields]]
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| [[Category:Conjectures]]
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Hello and welcome. My title is Irwin and I completely dig that name. North Dakota is her beginning place but she will have to move one working day or another. Body building is one of the issues I love most. She is a librarian but she's always wanted her own business.
Also visit my site; www.buzzbit.net