Smallest integer s > n such that 1 + 2s + 3s^2 + … + n s^n-1 is prime

Conjecture: For each n ≥ 2, there are infinitely many primes of the form 1 + 2s + … + n s^n-1 where s is a positive integer.

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-217785,
  title        = {Smallest integer s > n such that 1 + 2s + 3s^2 + … + n s^n-1 is prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-217785}},
  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: For each $n \geq 2$ , there are infinitely many primes of the form $1 + 2 s + \dots + n s^{n - 1}$ where $s$ is a positive integer.

conjecture_2. Conjecture: For each $n \geq 2$ , $a (n) < 12 n^{2}$ .

conjecture_3. Conjecture: The polynomials $s_{n} (x) = \sum_{k = 0}^{n} (k + 1) x^{k}$ for $n \geq 1$ are all irreducible over the field of rational numbers.

conjecture_4. Conjecture: For $n \geq 1$ , the polynomial $s_{n} (x) = \sum_{k = 0}^{n} (k + 1) x^{k}$ is reducible modulo every prime if and only if $n$ has the form $8 k (k + 1)$ where $k$ is a positive integer.

For $n \geq 2$ , $a (n)$ is the smallest integer $s > n$ such that the sum $P_{n} (s) = \sum_{k = 0}^{n - 1} (k + 1) s^{k}$ is a prime number.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«217785» (4 statements).

theorem conjecture_1 (n : ℕ) (hn : 2 ≤ n) :
    {s : ℕ | 0 < s ∧ Nat.Prime (polyP n s)}.Infinite

theorem conjecture_2 (n : ℕ) (hn : 2 ≤ n) :
    ∃ m < 12 * n ^ 2, a n = some m

theorem conjecture_3 (n : ℕ) (hn : 1 ≤ n) :
    Irreducible (Polynomial.map (Int.castRingHom ℚ) (polyS n))

theorem conjecture_4 (n : ℕ) (hn : 1 ≤ n) :
    (∀ p : ℕ, p.Prime → ¬ Irreducible (Polynomial.map (Int.castRingHom (ZMod p)) (polyS n))) ↔
      ∃ k : ℕ, 0 < k ∧ n = 8 * k * (k + 1)

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

A217785

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.