Smallest $m$ such that $n^3 + m^3 + 1$ is prime

formal-conjectures · oeis · ams-11

$a(n)$ is the smallest natural number $m \ge 1$ such that $n^3 + m^3 + 1$ is prime.

Conjecture 2: For any $k \ge 3$, there are infinitely many primes of the form $n^k + m^k + 1$
for $n, m \ge 1$.
- _Ulrich Krug_, 2009

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/A159829
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/OEIS/159829.lean
Local target: Corpus.OEIS159829.conjecture2
Source SHA-256: 77a70e94ccc2e947e9e163188b971d9a2056420907bebfb47f6a4030ba02f183
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