Supercongruences modulo p^(3k) and integrality for factorial ratios with fractional arguments, with proofs of conjectures of P. Bala
Year: 2026
Type: preprint
Abstract: Let a(N) = c ∏ Γ(α_i N + 1)^(e_i) with rational α_i > 0 of common denominator d, integers e_i and a rational constant c. Assume that Σ e_i α_i = 0 and that, for every N, the factors whose arguments α_i N have the same non-zero fractional part have total exponent zero. We prove that a(n p^k)/a(n p^(k−1)) ∈ 1 + p^(3k) Z_p for every prime p ≥ 5 not dividing d and all n, k ≥ 1, so that a(n p^k) ≡ a(n p^(k−1)) (mod p^(3k)) whenever a is p-integral. The proof is elementary: a decomposition into blocks of consecutive factors and Wolstenholme-type congruences for harmonic sums. We also write the p-adic valuation of a(N), for p ∤ d, as a sum of values of explicit step functions, and for prime d give a closed formula for the d-adic valuation; together they give a finite sufficient test for integrality. As consequences we prove the following conjectures of P. Bala in the OEIS: the supercongruences modulo p^(3k) for twenty-eight sequences — A275652, A275654, A347854–A347858, A352651, A352652, A364172–A364184, A364304, A364515–A364517, A365026, A365027 (proofs generated by language models had been reported for A364173 and A364176, the one for A364173 assuming integrality) —; the same supercongruences for every row of the square arrays A364303, A364518, A364519 and A365025 and for every row of A364513 except row 1 (A364514); for Bala's family in A352651, u_N(n) = C(Nn,2n) C(Nn/2,2n) C(2n,n)^2 / C(Nn/2,n)^2, the integrality and the supercongruences for every N ≥ 4 (and N = 2); and the integrality of A347854–A347858, A364172–A364174, A364176–A364184 and A276100–A276102 (the integrality of A364175 was proved by D. Radcliffe), and of A352652, which answers Bala's question in that entry. For integer arguments the supercongruences follow directly from the Jacobsthal–Kazandzidis congruence; we record that this settles the remarks "may hold" in A211417 and A295431 (for all n, and for the 52 sporadic integral factorial ratios) and the conjecture in A364506. We also correct two closed formulas recorded in A364303, which give T(n−1,k) instead of T(n,k). Row 1 of A364513 (A364514), the case N = 3 of A352651 (which is the same sequence up to sign) and the cases N ≤ −1 remain open. The statements were collected with the author's SyntheticMind engine from a local copy of the OEIS. The proofs and the text were prepared with the assistance of an AI system under the author's direction (including two rounds of a separate AI review of the proofs); the author checked every step and takes full responsibility for the content. Files: Blanco_Gomez_2026_factorial_ratios_fractional_EN.pdf is the note (8 pages); Blanco_Gomez_2026_cocientes_factoriales_fraccionarios_ES.pdf is the Spanish version (same content); verify_factorial_ratios.py is a verification script (Python 3, exact rational arithmetic, standard library only) that checks every formula against the OEIS data and the closed forms against the definitions, the balance conditions, the exact reduction identity, the lemmas, the supercongruences (more than 2500 cases, the exponent 3k attained), the valuation formula and the integrality certificates, the family u_N, the corrections to A364303, and controls (failure for p = 3); verify_factorial_ratios.log is its output (161 checks, ALL CHECKS PASSED); datos_oeis_cocientes.json contains the OEIS terms used.
Author: Blanco Gómez, Roberto
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222436079
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