Prime multiplicities in Bouras's continued fractions

M. Bouras, OEIS A363347, COMMENTS, Conjecture 3:

Conjecture 3: Except for 5, the primes all appear exactly twice.

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

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

For $n\ge3$, let $K(n,x)$ be the reciprocal of the right-folded fraction $2-3/(3-4/(\cdots- n/x))$. Define $a_4(n)=\operatorname{den}(K(n,-4))$ and $a_2(n)=\operatorname{den}(K(n,-2))$, using the positive reduced rational denominator. The first claim counts exactly two indices for every occurring prime $p\ne5$; the second counts exactly two or three for every occurring prime $p$. This dossier anchors A363347 Conjecture 3; the sibling settlement has its own dossier. Neither statement asserts that every prime occurs. This is the binding reading in preregistrations #18626 and #18627, both tier 1 external OEIS problems.

Statement echo

| Source clause | Formal counterpart | Fidelity | | --- | --- | --- | | Continued fraction, terminal value $-4$ | BourasA363347.a, using BourasA356360.K and nested | Literal reciprocal and right fold; Rat.den | | Continued fraction, terminal value $-2$ | BourasA363102.a, using the same K | Literal reciprocal and right fold; Rat.den | | Prime values | Nat.Prime and an existential occurrence guard | Includes precisely the primes taken by the sequence | | Except for $5$ | BourasA363347.claim has the guard $p\ne5$ | No exclusion is added to the second claim | | Indices start at three | Both occurrence sets include $3\le n$ | Matches the OEIS offsets | | Exactly twice | BourasA363347.claim uses Set.ncard = 2 | Its proof constructs a finite pair | | Twice or three times | BourasA363102.claim uses the disjunction of cardinalities two and three | Its proof constructs a pair or triple |

The natural polynomial BourasPolynomials.U is $U(n)=n^2-2$, with truncated natural subtraction. At $n\ge3$ it agrees with the integer polynomial. The source's rational fold expressions are represented by the existing frozen BourasA356360.nested and K declarations in this module. The public BourasA372763OnceRefutation.nested and K owners are textually equal; reusing them here would require changing or removing those frozen declarations, so both copies remain. No duplicate-retention authorization is claimed.

The results determine unbounded occurrence counts, rather than checking a finite prefix. A363102 uses the shared arithmetic and root-pair declarations exported by the A363347 module. In particular U, cast_U, U_prime_constraints, and U_prime_of_root are consumed by BourasA363102.result.

The literature checks recorded in #18626 and #18627 distinguish these questions from A363347 Conjectures 1 and 2 and A363102 Conjecture 1, which the OEIS entries and Adamczewski, arXiv:2608.11941, Table 1, describe as settled. Those earlier results are not on these modules' dependency paths. The recorded screen covers the OEIS entries, that article, the Epoch OEIS Open list, google-deepmind/formal-conjectures, and google-deepmind/alphaproof-nexus-results. It found no settlement of the two multiplicity claims in that scope. Literature completeness is ASSUMED-UNVERIFIED; this is no priority claim.

For a negative integer tail $x$, put $D=(n-1)x-n(n-2)$. A recurrence for the numerator gives a linear fractional expression for $K(n,x)$ and a factorial determinant identity. Their gcd consequence is $\operatorname{den}(K(n,x))=|D|/\gcd(|D|,n!/2)$. For $x=-4$, $|D|=(n+1)^2-5$; for $x=-2$, $|D|=n^2-2$. Factorial divisibility and elementary residue exclusions constrain a surviving prime denominator. The remaining roots modulo that prime form a pair. For $x=-2$ a possible extra index satisfies $n^2=2p^2+2$ and is unique.

A prime $p\ne5$ occurring in A363347 at a number of indices other than two, or an occurring prime in A363102 at a number of indices other than two or three, would contradict the corresponding claim. The occurrence guard and the lower bound on indices are part of both tests.

The settling declarations are D5.S3.Arith.Congruence.BourasA363347PrimeMultiplicity.BourasA363347.result and D5.S3.Arith.Congruence.BourasA363102PrimeMultiplicity.BourasA363102.result. Their Scribe theorem nodes display the complete claims. The first carries a Proved resolution for bouras; the second carries a Proved resolution for bouras-a363102.

Finite computation

Occurrence-count experiment. At that immutable revision, run python3 docs/reports/oeis-bouras-continued-fraction-occurrence-counts/check.py 300 20000 from the experiments repository. Exit code: 0. The SHA-256 of check.py is 05b533313a609bd3bf139aef952a062566625488521351d2d2955731427d9ffb. The script reports RESULT PASS. Direct rational-fraction checks cover $3\le n\le300$; the denominator-based occurrence enumeration covers $3\le n\le20000$ and primes $p\le10000$. A363347 has 609 primes counted twice, $5$ once at $n=4$, and 619 absent. A363102 has 600 primes counted twice, three primes counted three times $(p,n_{\rm extra})=(7,10),(41,58),(239,338)$, and 626 absent. These are finite computed readings, not a proof of a uniform extension.

Escape audit

The escape audit is unfinished for all seven public theorem targets: cast_U, root_fiber_eq_pair, square_residue_pair, U_prime_constraints, U_prime_of_root, and the two result declarations. Issue #21990 names their full source owners and the missing dependent source reconstruction, bridge, variation, sensitivity, and dependence evidence. No declared_validated registration is claimed.

What the settlement shows

1. Kernel-checked: BourasShared.denominator_formula_aux in the first module proves the displayed denominator formula for every $n\ge3$ and every negative integer tail. Its parameter is $m=n-3$; no fixed-tail restriction is imposed.
2. Proved in prose: for every occurring prime $p\ne5$ in A363347, its occurrence set is exactly two modular roots with indices summing to $p-2$. The first module's T_prime_constraints gives $n+1<p$ and $p\mid (n+1)^2-5$ at every occurrence, and T_prime_of_root gives the reverse implication. square_residue_pair at shift one and residue five constructs distinct $r,q\ge3$ with $r+q+2=p$; root_fiber_eq_pair identifies the complete fiber. Substituting the two implications gives precisely the occurrence set ${r,q}$. These are the intermediate steps used by BourasA363347.result; its exported statement asserts the cardinality.
3. Proved in prose, with kernel-checked necessary condition: for an occurring prime in A363102, the indices below $p$ form exactly a pair summing to $p$. Apply square_residue_pair at shift zero and residue two and use root_fiber_eq_pair; cast_U, U_prime_constraints, and U_prime_of_root identify that fiber with occurrences below $p$. For $n\ge p$, U_prime_constraints gives $n^2=2p^2+2$. If two natural indices satisfy this equation, their equal squares make them equal. This proves at most one extra index and yields the pair/triple construction in BourasA363102.result.
4. Computed: the prime-count distributions and the three extra-index pairs for $p\le10000$, $3\le n\le20000$ are exactly the finite readings in Evidence. The pinned script, its SHA-256, command, exit code, and rational check range are given there.
5. Open here: the converse that every prime Pell solution gives a third occurrence and the full Newman–Shanks–Williams equivalence are not formalized by these results. The exported count claims do not require them.
6. Open: infinitude of primes with a third occurrence.
7. Open here: the additional A363347 gcd pairing identity and general negative-tail occurrence classifications. The denominator formula alone does not classify all tails.

The decisive distinction is the possible square deficit after cancellation against $n!/2$: it is excluded for the $-4$ polynomial and permits the Pell-type exception for $-2$. The first occurrence-pair identity follows as in item 2; the additional gcd identity needs control of common small prime factors. The settlement supplies the two multiplicity conjectures alongside the source's previously settled prime-value statements. It asserts no new theorem about the source's other conjectures or the open infinitude question.

Literature completeness and historical priority are unverified beyond the recorded search scope. The experiment certifies only its finite ranges. Four-slot escape registrations remain unfinished under issue #21990; kernel proof and registration completion are separate assertions.