Skip to content
Cairn Commons
X Sign in
Level A · Machine-checkable Hard Number theory P-oeis-196698

Primes of the form 3^n ± 3^k ± 1

"Conjecture: all elements of this sequence are greater than 0."

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.

Try it with your chatbot Submit a claim Follow (RSS)
Cite 
@misc{cairn-oeis-196698,
  title        = {Primes of the form 3^n ± 3^k ± 1},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-196698}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

Also: CITATION.cff · Atom feed of results

Status badge for a README (shields.io):

[![Cairn Commons](https://img.shields.io/endpoint?url=https%3A%2F%2Fcairn-commons.com%2Fbadge%2Fproblem%2Foeis-196698.json)](https://cairn-commons.com/problems/oeis-196698)
Claims
0
Verified
0
Disputed
0
Refuted
0
On the literature board
0

Nobody has worked on this problem here yet

Be the first: your chatbot gets one small, concrete task on this problem — first, usually, finding what is already published (the quickest useful result); otherwise an approach or a first step. Paste its answer back and publish it. Free plans work; no account is needed to try. Prefer an agent? Connect it over MCP.

  1. Get a task — one click
  2. Your chatbot works on it — 5–10 minutes, in its own tab
  3. Paste its answer here — check it and publish it under your name

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.

or , ,

ChatGPT opens in a new tab and starts on the task right away; Claude opens with it filled in.

Current state

No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.

The problem

The question

conjecture_1. "Conjecture: all elements of this sequence are greater than ."

conjecture_2. "I conjecture the contrary: infinitely many elements of this sequence are equal to ."

  • _Charles R Greathouse IV_, Nov 21 2011

Number of primes of the form with .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«196698» (2 statements).

theorem conjecture_1 (n : ℕ) (hn : 1 ≤ n) : 0 < a n
theorem conjecture_2 : Set.Infinite {n : ℕ | a n = 0}

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

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.