Combinatorics

Ramanujan's sum

Central binomial coefficient

Gaussian binomial coefficient

Pure mathematics

Mathematics

Binomial coefficient

Congruence relation

Poisson distribution

Statistics

Prime (order theory)

Negative binomial distribution

Programming language

Computer science

Binomial (polynomial)

Integer (computer science)

Proceedings of the Royal Society of Edinburgh Section A Mathematics

We prove some congruences on sums involving fourth powers of central q -binomial coefficients. As a conclusion, we confirm the following supercongruence observed by Long [Pacific J. Math. 249 (2011), 405–418]: $$\sum\limits_{k = 0}^{((p^r-1)/(2))} {\displaystyle{{4k + 1} \over {{256}^k}}} \left( \matrix{2k \cr k} \right)^4\equiv p^r\quad \left( {\bmod p^{r + 3}} \right),$$ where p ⩾5 is a prime and r is a positive integer. Our method is similar to but a little different from the WZ method used by Zudilin to prove Ramanujan-type supercongruences.

Some congruences involving fourth powers of central <i>q</i>-binomial coefficients

Some congruences involving fourth powers of central q-binomial coefficients