## Functiones et Approximatio Commentarii Mathematici

### A note on the congruences with sums of powers of binomial coefficients

#### Abstract

Let $p\geq 7$ be a prime, $l\geq 0$ be an integer and $k,~m$ be two positive integers, we obtain the following congruences, $\sum\limits_{s=lp}^{(l+1)p-1}\binom{kp-1}{s}^m\equiv\begin{cases}\binom{k-1}{l}^m2^{km(p-1)}\pmod{p^3},&\text{if}~2\nmid m,\\ \binom{k-1}{l}^m\binom{kmp-2}{p-1}\pmod{p^{4}},&\text{if}~2\mid m; \end{cases}$ and $\sum\limits_{s=lp}^{(l+1)p-1}(-1)^s\binom{kp-1}{s}^m\equiv\begin{cases}(-1)^l\binom{k-1}{l}^m2^{km(p-1)}\pmod{p^3}, &\text{if}~2\mid m,\\ (-1)^l\binom{k-1}{l}^m\binom{kmp-2}{p-1}\pmod{p^{4}}, &\text{if}~2\nmid m. \end{cases}$ Let $p$ and $q$ are distinct odd primes and $k$ be a positive integer, we have $\binom{kpq-1}{(pq-1)/2}\equiv \binom{kp-1}{(p-1)/2}\binom{kq-1}{(q-1)/2}\pmod {pq}.$

#### Article information

Source
Funct. Approx. Comment. Math., Volume 58, Number 2 (2018), 221-232.

Dates
First available in Project Euclid: 2 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.facm/1512183763

Digital Object Identifier
doi:10.7169/facm/1694

Mathematical Reviews number (MathSciNet)
MR3816076

Zentralblatt MATH identifier
06924929

#### Citation

Shen, Zhongyan; Cai, Tianxin. A note on the congruences with sums of powers of binomial coefficients. Funct. Approx. Comment. Math. 58 (2018), no. 2, 221--232. doi:10.7169/facm/1694. https://projecteuclid.org/euclid.facm/1512183763

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