Undirected circular permutations with adjacent products minus one all prime

Conjecture: a(n) > 0 for all n > 5 with n not equal to 13.

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-oeis-229232,
  title        = {Undirected circular permutations with adjacent products minus one all prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-229232}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

conjecture_1. Conjecture: $a (n) > 0$ for all $n > 5$ with $n$ not equal to $13$ .

conjecture_2. Conjecture: For any integer $n > 1$ , there is a permutation $π (1), \dots, π (n)$ of $1, \dots, n$ such that the $n$ numbers $2 π (1) π (2) - 1, \dots, 2 π (n - 1) π (n) - 1, 2 π (n) π (1) - 1$ are all prime.

_Zhi-Wei Sun_

conjecture_3. Conjecture: For any positive integer $n$ not equal to $4$ , there is a permutation $π (1), \dots, π (n)$ of $1, \dots, n$ such that the $n$ numbers $2 π (1) π (2) + 1, \dots, 2 π (n - 1) π (n) + 1, 2 π (n) π (1) + 1$ are all prime.

_Zhi-Wei Sun_

conjecture_4. Conjecture: Let $F$ be a finite field with $q > 7$ elements. Then, there is a circular permutation $a_{1}, \dots, a_{q - 1}$ of the $q - 1$ nonzero elements of $F$ such that all the $q - 1$ elements $a_{1} a_{2} - 1, a_{2} a_{3} - 1, \dots, a_{q - 2} a_{q - 1} - 1, a_{q - 1} a_{1} - 1$ are primitive elements of the field $F$ (i.e., generators of the multiplicative group $F ∖ {0}$ ).

_Zhi-Wei Sun_

conjecture_5. Conjecture: Let $F$ be a finite field with $q > 7$ elements. There is a circular permutation $b_{1}, \dots, b_{q - 1}$ of the $q - 1$ nonzero elements of $F$ such that all the $q - 1$ elements $b_{1} b_{2} + 1, b_{2} b_{3} + 1, \dots, b_{q - 2} b_{q - 1} + 1, b_{q - 1} b_{1} + 1$ are primitive elements of the field $F$ (i.e., generators of the multiplicative group $F ∖ {0}$ ).

_Zhi-Wei Sun_

Number of undirected circular permutations $π (1), \dots, π (n)$ of $1, \dots, n$ with the $n$ numbers $π (1) π (2) - 1, π (2) π (3) - 1, \dots, π (n - 1) π (n) - 1, π (n) π (1) - 1$ all prime.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«229232» (5 statements).

theorem conjecture_1 (n : ℕ) (hn : n > 5) (hne : n ≠ 13) : a n > 0

theorem conjecture_2 (n : ℕ) (hn : n > 1) :
    ∃ π : Equiv.Perm (Fin n), ∀ i : Fin n,
      (2 * ((π i).val + 1) *
        ((π ⟨(i.val + 1) % n, Nat.mod_lt _ (by omega)⟩).val + 1) - 1).Prime

theorem conjecture_3 (n : ℕ) (hn : n > 0) (hne : n ≠ 4) :
    ∃ π : Equiv.Perm (Fin n), ∀ i : Fin n,
      (2 * ((π i).val + 1) * ((π ⟨(i.val + 1) % n, Nat.mod_lt _ hn⟩).val + 1) + 1).Prime

theorem conjecture_4 (F : Type*) [Field F] [Fintype F] (hF : Fintype.card F > 7) :
    let m := Fintype.card F - 1
    ∃ a : Fin m ≃ Fˣ,
      ∀ i : Fin m, IsPrimitiveRoot
        ((a i : F) * (a ⟨(i.val + 1) % m, Nat.mod_lt _ (by omega)⟩ : F) - 1) m

theorem conjecture_5 (F : Type*) [Field F] [Fintype F] (hF : Fintype.card F > 7) :
    let m := Fintype.card F - 1
    ∃ b : Fin m ≃ Fˣ,
      ∀ i : Fin m, IsPrimitiveRoot
        ((b i : F) * (b ⟨(i.val + 1) % m, Nat.mod_lt _ (by omega)⟩ : F) + 1) m

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

A229232

Source and licence

Imported from Formal Conjectures (OEIS), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.