slug: bouras-a363102
bibkey: bouras2023continuedfractionmultiplicity
doi: null
url: https://oeis.org/A363102
triage: theorem
motivation_gids:
  - D5/S3/Arith/Congruence/BourasA363102PrimeMultiplicity

Two or three prime occurrences in Bouras A363102

Problem

M. Bouras, OEIS A363102, COMMENTS, Conjecture 2:

Conjecture 2: All prime numbers appear either twice (same as A356247 and A357127) or three times.

For $n\ge 3$, $a(n)$ is the positive reduced denominator of the rational continued fraction $1/(2-3/(3-4/(\cdots -n/(-2))))$. The claim quantifies over primes that occur, as specified in preregistration #18627. It does not assert that every prime occurs. This dossier anchors only Conjecture 2; the $-4$ sequence has its own dossier.

Statement echo

Motivation

The result determines the occurrence count for every prime value, uniformly in the prime and in the index. It consumes the shared polynomial, cast, prime constraints, reverse-root theorem and root-pair theorems exported by BourasA363347PrimeMultiplicity.

Gap

The screen recorded in #18627 found no settlement in the OEIS entry, Adamczewski’s arXiv:2608.11941, the Epoch OEIS Open list, or the named Google DeepMind formal-conjecture and AlphaProof result collections. The source’s prime-value Conjecture 1 is already literature-attested; it is not a dependency of this module. Literature completeness is ASSUMED-UNVERIFIED.

Route

The shared denominator formula gives $a(n)=(n^2-2)/\gcd (n^2-2,n!/2)$ for $n\ge 3$. U_prime_constraints says that an occurring prime is at least seven, divides $n^2-2$, and either exceeds $n$ or satisfies $n^2=2p^2+2$. The two roots of $n^2\equiv 2(\mathrm{mod}p)$ below $p$ are paired by $n\mapsto p-n$. U_prime_of_root turns these two roots into occurrences. Equal natural squares make an index above $p$ unique, so the total is two or three.

Falsifier

An occurring prime whose full index set at $n\ge 3$ has cardinality other than two or three contradicts the literal claim. An absent prime is outside its occurrence hypothesis.

Evidence

D5.S3.Arith.Congruence.BourasA363102PrimeMultiplicity.BourasA363102.result proves the complete occurrence-count claim. The Scribe node displays its full proposition and carries a Proved resolution for this dossier.

The pinned experiment is run with python3 docs/reports/oeis-bouras-continued-fraction-occurrence-counts/check.py 300 20000 from the experiments repository; exit 0. Its script SHA-256 is 05b533313a609bd3bf139aef952a062566625488521351d2d2955731427d9ffb. The script reports RESULT PASS. Direct rational checks cover $3\le n\le 300$. Occurrence enumeration covers $3\le n\le 20000$ and $p\le 10000$: 600 primes occur twice, three occur three times, and 626 are absent. The triples have $(p,n_{\mathrm{extra}})$ equal to $(7,10),(41,58),(239,338)$.

The escape audit is unfinished for BourasA363102.result and the six public theorem targets in the shared module. Issue #21990 lists the missing source reconstruction and bridge/intervention evidence. No declared_validated registration is claimed.

Triage

What the settlement shows

• Kernel-checked: BourasA363102.result proves that every occurring prime has two or three indices. U_prime_constraints gives the necessary Pell-type equation $n^2=2p^2+2$ at each extra index.
• Proved in prose: square_residue_pair at shift zero and residue two, together with root_fiber_eq_pair, cast_U, and U_prime_of_root, gives the complete pair below $p$, with sum $p$. Any two extra indices have equal squares $2p^2+2$ and hence are equal. These are the intermediate statements used to construct the pair or triple in the settling proof.
• Computed: the exact finite distributions and third-index pairs are the experiment readings in Evidence, within its stated ranges.
• Open here: the converse from a prime Pell solution to a third occurrence and the complete Newman–Shanks–Williams equivalence are not formalized.
• Open: infinitude of primes with a third occurrence.
• The common negative-tail denominator identity, the A363347 root pairing, the finite distributions of both sequences, and the open gcd-pairing and general-tail classification questions are treated, with their evidence kinds, in the family Triage.

The square deficit after factorial cancellation explains why this sequence permits a third occurrence while the $-4$ sequence has only the root pair. The result settles the count claim alongside the previously known prime-value statement. It asserts no new result about infinitude or the source’s other conjectures.

ASSUMED-UNVERIFIED

Literature completeness and priority remain unverified beyond the recorded screen. The experiment has only finite scope. Escape registration is unfinished under #21990 and is distinct from the kernel-checked occurrence theorem.