Product of two consecutive primes modulo the next prime

Conjecture: For x > 10^9, the most frequent value in a(n), n=1… x, has form 120k.

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-182126,
  title        = {Product of two consecutive primes modulo the next prime},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-182126}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

conjecture1. Conjecture: For $x > 1 0^{9}$ , the most frequent value in $a (n)$ , $n = 1 \dots x$ , has form $120 k$ .

conjecture2. Let $b = prime (n + 2) - prime (n)$ and $c = prime (n + 2) - prime (n + 1)$ . Conjecture: for $n > 61$ , $a (n) = b \cdot c$ .

_Charles R Greathouse IV_, May 11 2012

conjecture3. Are 2, 7, 11, 13, 29 the only primes in this sequence?

_Hugo Pfoertner_, Sep 22 2025

The sequence is defined by $a (n) = prime (n) \cdot prime (n + 1) mod prime (n + 2),$ where $prime (k)$ is the $k$ -th prime number ( $prime (1) = 2$ ).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«182126» (3 statements).

theorem conjecture1 (x : ℕ) (hx : 10 ^ 9 < x) (v₀ : ℕ) (hv : IsMostFrequent x v₀) :
    120 ∣ v₀

theorem conjecture2 (n : ℕ) (hn : 61 < n) :
    let b := prime (n + 2) - prime n
    let c := prime (n + 2) - prime (n + 1)
    a n = b * c

theorem conjecture3 (n : ℕ) (hn : 0 < n) :
    (a n).Prime ↔ a n ∈ ([2, 7, 11, 13, 29] : List ℕ)

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

A182126

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.