Supercongruences for Binomial Coefficients C(np^k,mp^k): Proving the Conjecture in Remark 6 of Helou‑Terjanian (2008)

Contact: Hexianxuan He Email: [email protected] 联系邮箱: [email protected] Hexianxuan Second Combinatorial Number Theorem 何先玄组合数第二定理 We establish supercongruences for binomial coefficients C(np^k,mp^k) for arbitrary integers k ≥ 1 and prime p ≥ 5. For the special case k=1, we prove the congruence which was posed as a conjecture in Remark 6 of Helou‑Terjanian (2008): C(np, mp) ≡ C(n, m) mod p^{3+v_p(nm(n−m)C(n,m))}. Furthermore, we obtain a recursive supercongruence relating C(np^k,mp^k) and C(np^{k−1},mp^{k−1}). These results generalize the classical theorems of Jacobsthal and Helou‑Terjanian, which are restricted to k=1. Potential applications to p‑adic lifts of quantum invariants in quantum topology are discussed. https://orcid.org/0009-0005-1999-250X References: He, H. (2026). A Complementary Refinement to the Helou–Terjanian Congruence, with an Elementary Proof of Wolstenholme's Theorem. Zenodo. https://doi.org/10.5281/zenodo.22127191