Number of primes p < n with prime(p)^2 + (prime(n)

Number of primes p < n with prime(p)^2 + (prime(n) - 1)^2 prime

Conjecture (i): a(n) > 0 unless n divides 6.

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-238585,
  title        = {Number of primes p < n with prime(p)^2 + (prime(n) - 1)^2 prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-238585}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-11}
}

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Claims

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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 (i): $a (n) > 0$ unless $n$ divides $6$ .

conjecture_2. Conjecture (i): $a (n) = 1$ only for $n \in {4, 5, 7, 10, 11, 12, 19, 21, 22, 31, 42, 44}$ .

conjecture_3. Conjecture (ii): If $n > 2$ is not equal to $9$ , then $prime (n)^{2} + (prime (p) - 1)^{2}$ is prime for some prime $p < n$ .

conjecture_4. Conjecture (iii), part 1: For $n > 3$ , there is a prime $p < n$ with $prime (p) + prime (n) + 1$ prime.

conjecture_5. Conjecture (iii), part 2: If $n > 9$ is not equal to $18$ , then $prime (p)^{2} + prime (n)^{2} - 1$ is prime for some prime $p < n$ .

The sequence $a (n)$ counts the number of primes $p < n$ such that $prime (p)^{2} + (prime (n) - 1)^{2}$ is prime, where $prime (i)$ denotes the $i$ -th prime number (1-indexed).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«238585» (5 statements).

theorem conjecture_1 (n : ℕ) (hn : 0 < n) :
    0 < a n ↔ ¬(n ∣ 6)

theorem conjecture_2 (n : ℕ) (hn : 0 < n) :
    a n = 1 ↔ n ∈ ({4, 5, 7, 10, 11, 12, 19, 21, 22, 31, 42, 44} : Set ℕ)

theorem conjecture_3 (n : ℕ) (hn : 2 < n) (hn9 : n ≠ 9) :
    ∃ p : ℕ, p < n ∧ p.Prime ∧
      Nat.Prime (Nat.nth Nat.Prime (n - 1) ^ 2 + (Nat.nth Nat.Prime (p - 1) - 1) ^ 2)

theorem conjecture_4 (n : ℕ) (hn : 3 < n) :
    ∃ p : ℕ, p < n ∧ p.Prime ∧
      Nat.Prime (Nat.nth Nat.Prime (p - 1) + Nat.nth Nat.Prime (n - 1) + 1)

theorem conjecture_5 (n : ℕ) (hn : 9 < n) (hn18 : n ≠ 18) :
    ∃ p : ℕ, p < n ∧ p.Prime ∧
      Nat.Prime (Nat.nth Nat.Prime (p - 1) ^ 2 + Nat.nth Nat.Prime (n - 1) ^ 2 - 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

A238585
Problems on combinatorial properties of primes by Zhi-Wei Sun (2014)

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.