Least $k > 0$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime

formal-conjectures · oeis · ams-11

The sequence $a(n)$ is the least positive integer $k$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is
prime,
if such $k$ exists; otherwise $a(n) = 0$.

$(k+1)(k+2)(k+3)(k+4) + 1 = (k^2 + 5k + 5)^2$, which is never prime. Hence $a(4) = 0$.
Conjecture: $a(n) = 0$ if and only if $n = 4$.

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/A078729
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/78729.lean
Local target: Corpus.OEIS78729.conjecture
Source SHA-256: 0e70beb10d6a9e316bc8510c4f5f78749e633fe75cb41bf73e93968b763f74c0
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.

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