Functiones et Approximatio Commentarii Mathematici

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

Tianxin Cai and Zhongyan Shen

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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

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

First available in Project Euclid: 2 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A07: Congruences; primitive roots; residue systems
Secondary: 11B65: Binomial coefficients; factorials; $q$-identities [See also 05A10, 05A30]

binomial coefficients prime powers congruences


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.

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