The conjecture was first made by Ernst Kummer on 28 December 1849 and 24 April 1853 in letters to Leopold Kronecker, reprinted in (Kummer 1975, pages 84, 93, 123–124), and independently rediscovered around 1920 by Philipp Furtwängler and Harry Vandiver (1946, p. 576), 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.
The first factor h1 is well understood and can be computed easily in terms of Bernoulli numbers, and is usually rather large.
Joe Buhler, Richard Crandall, and Reijo Ernvall et al. (2001) verified it for p < 12 million.
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.
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.
Mihăilescu (2010) gave a refined version of Washington's heuristic argument, suggesting that the Kummer–Vandiver conjecture is probably true.
Kurihara (1992) showed that the conjecture is equivalent to a statement in the algebraic K-theory of the integers, namely that Kn(Z) = 0 whenever n is a multiple of 4.