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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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

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

Keywords
binomial coefficients prime powers congruences

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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References

  • \labelCTX T.X. Cai, A. Granville, On the residues of binomial coefficients and their products modulo primes powers, Acta Mathenatica Sinica 18(2) (2002), 277–288.
  • \labelLe E. Lehmer, On congruences involving Bernoulli numbers and quotients of Fermat and Wilson, Ann. Math. 39 (1938), 350–360.
  • \labelLu E. Lucas, Amer. Jour. Mah. 1 (1879), 229–230.
  • \labelMo F. Morley, Note on the congruence $2^{4n}\equiv (-1)^n(2n)!/(n!)^2$, where $2n+1$ is a prime, Annals of Math. 9, (1895), 168–170.
  • \labelSZH Z.H. Sun, Congruences concerning Bernoulli numbers and Bernoulli polynomials, Discrete Appl. Math. 105 (2000), 193–223.
  • \labelTZ R. Tauraso, J.Q. Zhao, Congruences of alternating multiple harmonic sums,.
  • \labelWJ J. Wolstenholme, On certain properties of prime numbers, The Quarterly Journal of Pure and Applied Mathematics 5 (1862), 35–39.
  • \labelZhao J.Q. Zhao, Wolstenholme type Theorem for multiple harmonic sums, Int. J. Number Theory 4(1) (2008), 73–106.