OP-00009 Open

The Feit–Thompson conjecture on primes

There are no distinct primes $p$ and $q$ such that $\frac{q^p - 1}{q - 1}$ divides $\frac{p^q - 1}{p - 1}$

Source: W. Feit and J. G. Thompson, A solvability criterion for finite groups and some consequences, Proc. Natl. Acad. Sci. USA 48 (1962), 968–970.

Classification

Primary: NT — Number Theory
Secondary: GR
Keywords: divisibility

References

@article{feitthompson1962,
  author = {Feit, Walter and Thompson, John G.},
  title = {A solvability criterion for finite groups and some consequences},
  journal = {Proceedings of the National Academy of Sciences USA},
  volume = {48},
  pages = {968--970},
  year = {1962}
}

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

@misc{formalconjectures,
  author = {{The Formal Conjectures Authors}},
  title = {Formal Conjectures},
  url = {https://github.com/google-deepmind/formal-conjectures/blob/0771383387505c96d1b2f6a3d35088ad00892c5c/FormalConjectures/Wikipedia/FeitThompsonPrimeConjecture.lean#L30},
  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_00009.

/-- OP-00009: The Feit–Thompson conjecture on primes -/
def Problem_OP_00009 : Prop :=
  ∀ (p q : ℕ), Nat.Prime p → Nat.Prime q → p < q → ¬(q ^ p - 1) / (q - 1) ∣ (p ^ q - 1) / (p - 1)

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_00009

theorem resolution : RegistryProblems.Problem_OP_00009 := by
  sorry

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

[[statements]]
label = "Theorem 1"
lean = "resolution"
latex = "..."
problem = "OP-00009"
claim = "full"

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

theorem disproof : ¬ RegistryProblems.Problem_OP_00009 := 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-00009, "The Feit–Thompson conjecture on primes", clopen.solutions.