Year: 2026
Type: preprint
Abstract: Let $C_k=\binom{2k}{k}/(k+1)$ be the $k$-th Catalan number, and let $p$ be an odd prime. We establish exact finite-sum identities that, together with known binomial supercongruences and reductions of Sun, yield $\sum_{k=0}^{(p-1)/2}\frac{C_k^3}{16^k}\equiv2-24y^2-\frac{5p^2}{2y^2}\pmod{p^3}$ when $p=x^2+3y^2\equiv1\pmod3$, and $\sum_{k=0}^{(p-1)/2}\frac{C_k^3}{(-8)^k}\equiv-1-48y^2-\frac{33p^2}{16y^2}\pmod{p^3}$ when $p=x^2+4y^2\equiv1\pmod4$. We also evaluate the corresponding sum with denominator $16^k(k+1)^2$, and obtain two weighted Catalan supercongruences. The finite identities follow from the ratios of consecutive summands, symmetry of double sums, and telescoping. All endpoint terms are retained over the rational numbers before any modular reduction. In each bilinear identity, one endpoint denominator equals $p$; its cancellation produces the necessary term of order $p^2$. The precise arithmetic inputs are stated separately.
Source: arXiv (Cornell University)
Author: Liu, Yuqi
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7223883832