The complete first growth assertion of OEIS A222759

T. D. Noe, OEIS A222759, revision 12, November 2, 2024. The NAME is:

Conjectured number of primes p for which binomial(n*p,p) (mod p^3) does not equal n.

The entire first COMMENT is:

It appears that, for k > 2 and n >= prime(prime(k)^3), then a(n) >= k.

Let $P(j)$ be the one-based $j$-th prime for positive $j$, with $P(1)=2$. For every integer $k\ge3$ and every integer $n\ge1$ with $n\ge P(P(k)^3)$, the event is the existence of $k$ pairwise distinct positive primes $q_1,\ldots,q_k$ such that $\binom{nq_i}{q_i}\bmod q_i^3\ne n$ for every $i$. The remainder is the least nonnegative integer remainder, compared with $n$ itself. The threshold includes equality and cubes $P(k)$ inside the outer prime index. The event asserts at least $k$ witnesses without assuming finiteness of the entire set satisfying the predicate.

The formal domain is exactly the positive integer domain represented by naturals with $3\le k$ and $1\le n$. An injection $q:\mathrm{Fin}(k)\to \mathbb N$ supplies distinct witnesses, each explicitly positive and prime. The total extension $P(0)=2$ is unused on the event's indices; the natural remainder convention at modulus zero is unused on prime witnesses. The source identity and operator correspondence are in Library/Factorization/noe2013a222759.md.

The 2013 first-comment growth assertion is a Tier 1 published OEIS conjectural assertion. Its full unbounded event is settled by the literal remainder range and the sizes of the first $k$ primes. The frozen result provides the exact source event rather than a finite sample or an assumption about the global count.

The original preregistration is comment 6618295321. Its unchanged source event was registered at 2026-11-03T23:38:27Z, before execution at 2026-11-03T23:38:45Z; its deadline is 2026-11-11T00:00:00Z. These are supplied preregistration readings.

The bounded source assessment identifies revision 12's first COMMENT as still conjectural and finds no full-resolution citation in that entry. At repository revision 6edb23f4fc2d44228fd765e0a6c2f57e85f7532b, bounded identifier and statement-shape searches of D5, Library, Problems and theory found only this unit and its source note for the exact event. Pinned Mathlib searches found the arithmetic suppliers, without a named supplier of this entire OEIS assertion. The accepted source/history/literature assessment is fallible supplied evidence. These readings do not certify worldwide literature absence, originality or priority.

Set $m=P(k)^3>0$. The pinned prime-enumeration bound Nat.add_two_le_nth_prime at $m-1$ gives $m+1\le P(m)$. Consequently the source threshold implies $P(k)^3<n$. For $i\in\mathrm{Fin}(k)$ choose $q(i)=\operatorname{nth}(\mathrm{Prime},i)$, the first $k$ primes. Prime enumeration is injective and monotone, so these are distinct positive primes and $q(i)\le P(k)$. Thus $q(i)^3\le P(k)^3<n$.

Nat.mod_lt gives $\binom{nq(i)}{q(i)}\bmod q(i)^3<q(i)^3<n$. The literal remainder therefore differs from $n$. This is the ordinary remainder-range mechanism; it uses no special binomial congruence. Direct pinned suppliers include Nat.nth, Nat.prime_nth_prime, Nat.add_two_le_nth_prime, Nat.nth_injective, Nat.nth_monotone, Nat.infinite_setOfPred_prime, Nat.Prime.pos, pow_pos, pow_le_pow_left', Nat.mod_lt, Fin.ext and ne_of_lt.

A counterexample to this exact event would be integers $k\ge3$ and $n\ge P(P(k)^3)$ with fewer than $k$ positive prime witnesses to the literal remainder inequality. Such a counterexample contradicts the compiled result. A different placement of the cube, a congruence in place of literal remainder equality, or an added whole-set finiteness premise would break source fidelity. An earlier published resolution of this exact assertion would invalidate the external-open-problem eligibility claim without changing the verified arithmetic theorem.

The mathematical source is D5/S3/Arith/PrimeRemainders/OeisA222759Growth.lean. Its two definitions P and cubicRemainder are consumed by its sole public theorem result. The full result has statement identity sha256:2a4b8f3d982aa14d26dbee8dfc3e8292ceaea402c369fa3c37126a82b8dbb72c. Its full telescope retains $k,n$, both lower bounds and the nested prime threshold. There is no extra D5 companion declaration.

The per-theorem classification is proof_shape: bind-only, escape_witness: none, admission_basis: open-problem-resolution. The two earning definitions have no proof-shape claim. There are no direct previously frozen project dependencies; the arithmetic suppliers are pinned Mathlib and Lean. All three public declarations have axiom closure contained in propext, Classical.choice, Quot.sound, without sorry or an added axiom. The unbounded symbolic theorem has utility: none.

The canonical module membership record is Golden/Frozen/state/D5/S3/Arith/PrimeRemainders/OeisA222759Growth.lean.json. The matching Blueprint Scribe attaches one OpenProblemResolutionClaim with ResolutionKind.Proved to this result and this dossier's slug. The retained Reg mirror reconstructs the full original telescope through an Iff.rfl source-equivalence bridge. It supplies the actual law, whole-family rejection and sensitivity, and actual readout dependence. Its unknown continuation records a residual only; it asserts neither undecidability nor inexhaustibility.

• Proved: the entire first COMMENT event for every $k\ge3$ and every positive $n\ge P(P(k)^3)$, including the threshold endpoint. The evidence is the full compiled OeisA222759Growth.result, with an injection of $k$ positive prime witnesses and the literal remainder predicate.
• Proved within that result: the first $k$ primes have cubes below $n$; the ordinary remainder bound forces the required inequalities. This identifies the decisive mechanism, without claiming a new general binomial theorem or a new proof technique.
• Open here: whether a smaller threshold is optimal and what exact counts hold. The result does not establish the finiteness of the entire prime set, any exact value of $a(n)$, or the sample table.
• Outside this settlement: the second COMMENT and A222760. No conclusion about either follows as a credited result in this dossier. No dependency of those assertions on the first COMMENT has been verified here.

The supplied bounded prior and literature assessment is not a global semantic search or an exhaustive exclusion of earlier full resolutions. No independent priority, novelty, authorship-credit or external completion certificate is asserted. The ordinary proof mechanism is known arithmetic. The exact event above is distinguished from other comments, exact counts, whole-set finiteness and A222760. Those excluded questions remain unsettled by this delivery. Repository-wide required checks and final independent review are separate from the scoped local evidence.