Connections between binomial coefficients and binary quadratic forms
- Resource Type
- Original Paper
- Authors
- Mao, Guo-Shuai
- Source
- The Ramanujan Journal: An International Journal Devoted to the Areas of Mathematics Influenced by Ramanujan. :1-20
- Subject
- Congruences
Binomial coefficients
Harmonic numbers
p-adic Gamma function
Gauss and Jacobi sums
Binary quadratic forms
Primary 11A07
Secondary 05A10
11B68
11E16
- Language
- English
- ISSN
- 1382-4090
1572-9303
In this paper, we mainly prove some congruences involving binomial coefficients and binary quadratic forms. One such example is the following: Let p b be a prime such that p=x2+2y2≡1(mod8)p∑k=0p-12kk2(8k+1)16k≡3p∑k=0p-12kk2(8k+3)16k≡4x2-2p-p24x2(modp3).. Then, p=x2+2y2≡1(mod8)p∑k=0p-12kk2(8k+1)16k≡3p∑k=0p-12kk2(8k+3)16k≡4x2-2p-p24x2(modp3).