A Complementary Refinement to the Helou–Terjanian Congruence, with an Elementary Proof of Wolstenholme's Theorem

Hexianxuan First Combinatorial Number Theorem 何先玄组合数第一定理 In this paper, we first give an elementary combinatorial proof of Wolstenholme’s theorem. Furthermore, by adopting elementary combinatorial methods and p‑adic valuation analysis, we establish a refined and complementary higher‑order congruence for binomial coefficients. The congruence says that the binomial coefficient C(np, mp) is congruent to C(n, m) modulo p^(3 + v_p(n * C(n, m))). Here the binomial coefficient is given by C(n,m) = n! / (m! (n‑m)!). For comparison, the classical congruence due to Helou and Terjanian (2008) gives a similar congruence with a different valuation term. https://orcid.org/0009-0005-1999-250X https://doi.org/10.5281/zenodo.21333058 Hexianxuan Primality Test 何先玄素性检测