Selfridge's conjectures

PSW conjecture (Selfridge's test) Let p be an odd number, with p ≡ ± 2 pmod5, 2^p-1 ≡ 1 pmodp and F_p+1 ≡ 0 pmodp, then p is a prime number.

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.

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Cite

@misc{cairn-selfridge,
  title        = {Selfridge's conjectures},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/selfridge}},
  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

selfridge_conjecture. PSW conjecture (Selfridge's test) Let $p$ be an odd number, with $p \equiv \pm 2 (mod 5)$ , $2^{p - 1} \equiv 1 (mod p)$ and $F_{p + 1} \equiv 0 (mod p)$ , then $p$ is a prime number.

selfridge_seq_conjecture. Selfridge conjectured that the number of prime factors of the n-th Fermat number does not grow monotonically in $n$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Wikipedia.Selfridge (2 statements).

theorem selfridge_conjecture (p : ℕ) (hp : IsSelfridge p) : p.Prime

theorem selfridge_seq_conjecture : ¬ Monotone fermatFactors

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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.