Feit-Thompson conjecture on primes

There are no distinct primes p and q such that q^p - 1/q - 1 divides p^q - 1/p - 1

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-feit-thompson-prime-conjecture,
  title        = {Feit-Thompson conjecture on primes},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/feit-thompson-prime-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

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

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.FeitThompsonPrimeConjecture.

theorem feit_thompson_primes (p q : ℕ) (hp : p.Prime) (hq : q.Prime) (h : p < q) :
    ¬ (q ^ p - 1) / (q - 1) ∣ (p ^ q - 1) / (p - 1)

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Wikipedia

Source and licence

Imported from Formal Conjectures (Wikipedia), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.