The Kummer-Vandiver conjecture, also called the Vandiver conjecture, states that a prime p does not divide the class number of the maximal real subfield of the p-th cyclotomic field. Ernst Kummer first formulated the idea in letters to Leopold Kronecker in 1849 and 1853, and it was independently rediscovered around 1920 by Philipp Furtwängler and Harry Vandiver, whose name has become the conjecture's common label.
Facts
StatementThe conjecture states that a prime p does not divide the class number of the maximal real subfield of the p-th cyclotomic field. 1 Proposed YearFirst stated by Kummer in letters of 28 December 1849 and 24 April 1853; independently rediscovered around 1920 by Furtwangler and Vandiver. Progress Toward ResolutionComputational verification has been extended to all primes less than 2^31 with no counterexample found. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Kummer-Vandiver conjecture (Wikipedia)
Posed By
Source Kummer-Vandiver conjecture (Wikipedia)
Sources
1. Kummer-Vandiver Conjecture (Wikipedia)
Wikimedia FoundationLead section
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.
Lead section
The conjecture was first made by Ernst Kummer, on 28 December 1849 and 24 April 1853 in letters to Leopold Kronecker, and independently rediscovered around 1920 by Philipp Furtwängler and Harry Vandiver.
Opening section, before Background heading
The conjecture was first made by Ernst Kummer, on 28 December 1849 and 24 April 1853 in letters to Leopold Kronecker, and independently rediscovered around 1920 by Philipp Furtwängler and Harry Vandiver.
Evidence for and against the Kummer-Vandiver conjecture
Buhler and Harvey extended this to primes less than 163000000, and Hart, Harvey, and Ong extended this to primes less than 231.
View the Source Kummer-Vandiver conjecture (Wikipedia)
Posed By: Ernst Kummer, Lead paragraph
The conjecture was first made by Ernst Kummer, on 28 December 1849 and 24 April 1853 in letters to Leopold Kronecker, and independently rediscovered around 1920 by Philipp Furtwängler and Harry Vandiver.
In Branch: Number Theory, Lead section
states that a prime
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.