A+ CATEGORY SCIENTIFIC UNIT

# Publishing house / Journals and Serials / Acta Arithmetica / All issues

## Acta Arithmetica

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

## Supercongruences and binary quadratic forms

### Volume 199 / 2021

Acta Arithmetica 199 (2021), 1-32 MSC: Primary 11A07; Secondary 05A19, 11B65, 11E25. DOI: 10.4064/aa200308-27-9 Published online: 12 March 2021

#### Abstract

Let $p \gt 3$ be a prime, and let $a,b$ be two rational $p$-adic integers. We present general congruences for $\sum _{k=0}^{p-1}\binom ak\binom {-1-a}k\frac p{k+b}\pmod {p^2}$. Let $\{D_n\}$ be the Domb numbers given by $D_n=\sum _{k=0}^n\binom nk^2\binom {2k}k\binom {2n-2k}{n-k}$. We also prove that $$\sum _{n=0}^{p-1}\frac {D_n}{16^n}\equiv \sum _{n=0}^{p-1}\frac {D_n}{4^n}\equiv \begin {cases} 4x^2-2p\pmod {p^2} &\text {if 3\mid p-1 and so p=x^2+3y^2,}\\ 0\pmod {p^2} &\text {if p\equiv 2\pmod 3,}\end {cases}$$ which was conjectured by Z. W. Sun.

#### Authors

• Zhi-Hong SunSchool of Mathematics and Statistics
Huaiyin Normal University
Huaian, Jiangsu 223300, P.R. China
http://maths.hytc.edu.cn/szh1.htm
e-mail

## Search for IMPAN publications

Query phrase too short. Type at least 4 characters.