Product of two consecutive primes modulo the next prime

formal-conjectures · oeis · ams-11

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

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

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://oeis.org/A182126
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/182126.lean
Local target: Corpus.OEIS182126.conjecture1
Source SHA-256: d01858e30978c3034d9201a183871735e2de78a1a4d4780d1f66675999f93a84
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record