In mathematics, the Kummer–Vandiver conjecture, or Vandiver conjecture, states that a prime
p
{\displaystyle p}
does not divide the class number
h
K
{\displaystyle h_{K}}
of the maximal real subfield
K
=
Q
(
ζ
p
)
+
{\displaystyle K=\mathbb {Q} (\zeta _{p})^{+}}
of the
p
{\displaystyle p}
-th cyclotomic field. 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.
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.
Contents
Background
The class number
h
{\displaystyle h}
of the cyclotomic field
Q
(
ζ
p
)
{\displaystyle \mathbb {Q} (\zeta _{p})}
is a product of two integers
h
1
{\displaystyle h_{1}}
and
h
2
{\displaystyle h_{2}}
, called the first and second factors of the class number, where
h
2
{\displaystyle h_{2}}
is the class number of the maximal real subfield
K
=
Q
(
ζ
p
)
+
Evidence for and against the Kummer–Vandiver conjecture
Kummer verified the Kummer–Vandiver conjecture for
p
{\displaystyle p}
less than 200, and Vandiver extended this to
p
{\displaystyle p}
less than 600. Buhler, Crandall et al. verified it for p < 12000000. Buhler and Harvey extended this to primes less than 163000000, and Hart, Harvey, and Ong extended this to primes less than 231.
Washington describes an informal probability argument, based on rather dubious assumptions about the equidistribution of class numbers modulo
p
{\displaystyle p}
, suggesting that the number of primes less than
x
{\displaystyle x}
that are exceptions to the Kummer–Vandiver conjecture might grow like
(
log
log
x
)
/
2
{\displaystyle (\log \log x)/2}
Consequences of the Kummer–Vandiver conjecture
Kurihara showed that the conjecture is equivalent to a statement in the algebraic K-theory of the integers, namely that
K
n
(
Z
)
=
0
{\displaystyle K_{n}(\mathbb {Z} )=0}
whenever
n
{\displaystyle n}
is a multiple of
4
{\displaystyle 4}
. In fact, from the Kummer–Vandiver conjecture and the norm residue isomorphism theorem follows a full conjectural calculation of the
K
{\displaystyle K}
-groups for all values of
n
{\displaystyle n}
; see Quillen–Lichtenbaum conjecture for details.