OP-00007 Open

The Kummer–Vandiver conjecture

For every prime $p$, the class number of $\mathbb{Q}(\zeta_p)^+ = \mathbb{Q}(\zeta_p + \zeta_p^{-1})$, the maximal real subfield of the $p$-th cyclotomic field, is not divisible by $p$.

Source: Kummer, letters to Kronecker (1849, 1853); rediscovered by Vandiver, Amer. Math. Monthly 53 (1946). See Washington, Introduction to Cyclotomic Fields, §8.3.

Classification

Primary: NT — Number Theory
Keywords: class numbers, cyclotomic fields

References

@article{vandiver1946,
  author = {Vandiver, H. S.},
  title = {Fermat's last theorem: its history and the nature of the known results concerning it},
  journal = {American Mathematical Monthly},
  volume = {53},
  pages = {555--578},
  year = {1946}
}

@book{washington1997,
  author = {Washington, Lawrence C.},
  title = {Introduction to Cyclotomic Fields},
  edition = {2},
  series = {Graduate Texts in Mathematics},
  volume = {83},
  publisher = {Springer},
  year = {1997}
}

@article{hartharveyong2017,
  author = {Hart, William and Harvey, David and Ong, Wilson},
  title = {Irregular primes to two billion},
  journal = {Mathematics of Computation},
  volume = {86},
  pages = {3031--3049},
  year = {2017}
}

@misc{wikipedia,
  title = {Kummer–Vandiver conjecture},
  howpublished = {Wikipedia},
  url = {https://en.wikipedia.org/wiki/Kummer%E2%80%93Vandiver_conjecture}
}

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/KummerVandiver.lean#L34},
  note = {Lean statement under Apache-2.0; problem text under CC BY 4.0, or CC BY-SA 4.0 where it is based on Wikipedia}
}

Canonical Lean statement

A submission resolves this problem by proving the constant Problem_OP_00007.

/-- OP-00007: The Kummer–Vandiver conjecture -/
def Problem_OP_00007 : Prop :=
  ∀ (p : ℕ+),
      p.Prime → ¬↑p ∣ NumberField.classNumber ↥(NumberField.maximalRealSubfield (CyclotomicField ↑p ℚ))

Formalised by The Formal Conjectures Authors (Apache-2.0); reviewed by clemens-admin.

Available to match against in: Lean v4.33.1 with Mathlib v4.33.1.

Resolving this problem in Lean

Add the problems package as a dependency, import this problem's module, and prove the constant. Any published revision works.

In lakefile.toml:

[[require]]
name = "RegistryProblems"
git = "https://github.com/cthalhammer/registry-problems"
rev = "<published revision>"

In your proof:

import RegistryProblems.OP_00007

theorem resolution : RegistryProblems.Problem_OP_00007 := by
  sorry

In preprint.toml, mapping the claim to the declaration that establishes it:

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00007"
claim = "full"

To disprove it instead, prove the negation and claim "disproof", or "counterexample" if the proof exhibits one:

theorem disproof : ¬ RegistryProblems.Problem_OP_00007 := by
  sorry

You can also write the statement out in full; the verifier checks it matches up to definitional equality. Partial results and reductions are not matched against this statement.

Citing this problem

This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.

OP-00007, "The Kummer–Vandiver conjecture", clopen.solutions.