P-Adic Binomial Coefficients MOD P

1985 ◽  
Vol 92 (8) ◽  
pp. 576 ◽  
Author(s):  
Roger C. Alperin
Keyword(s):  
Binomial Coefficients ◽  
Mod P

P-Adic Binomial Coefficients Mod P

1985 ◽  
Vol 92 (8) ◽  
pp. 576-578 ◽  
Author(s):  
Roger C. Alperin
Keyword(s):  
Binomial Coefficients ◽  
Mod P

scholarly journals Supercongruences involving Apéry-like numbers and binomial coefficients

2021 ◽  
Vol 7 (2) ◽  
pp. 2729-2781
Author(s):  
Zhi-Hong Sun ◽  
Keyword(s):  
Binomial Coefficients ◽  
Mod P

<abstract><p>Let $ \{S_n\} $ be the Apéry-like sequence given by $ S_n = \sum_{k = 0}^n\binom nk\binom{2k}k\binom{2n-2k}{n-k} $. We show that for any odd prime $ p $, $ \sum_{n = 1}^{p-1}\frac {nS_n}{8^n}{\equiv} (1-(-1)^{\frac{p-1}2})p^2\ (\text{ mod}\ {p^3}) $. Let $ \{Q_n\} $ be the Apéry-like sequence given by $ Q_n = \sum_{k = 0}^n\binom nk(-8)^{n-k}\sum_{r = 0}^k\binom kr^3 $. We establish many congruences concerning $ Q_n $. For an odd prime $ p $, we also deduce congruences for $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k}\ (\text{ mod}\ {p^3}) $, $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(k+1)^2}\ (\text{ mod}\ {p^2}) $ and $ \sum_{k = 0}^{p-1}\binom{2k}k^3\frac 1{64^k(2k-1)}\ (\text{ mod}\ p) $, and pose lots of conjectures on congruences involving binomial coefficients and Apéry-like numbers.</p></abstract>

NEW SUPERCONGRUENCES INVOLVING PRODUCTS OF TWO BINOMIAL COEFFICIENTS

2019 ◽  
Vol 101 (3) ◽  
pp. 367-378
Author(s):  
ZHI-HONG SUN
Keyword(s):  
Binomial Coefficients ◽  
Euler Polynomials ◽  
Image Position ◽  
Mod P

Let $p>3$ be a prime and let $a$ be a rational $p$-adic integer with $a\not \equiv 0\;(\text{mod}\;p)$. We evaluate $$\begin{eqnarray}\mathop{\sum }_{k=1}^{(p-1)/2}\frac{1}{k}\binom{a}{k}\binom{-1-a}{k}\quad \text{and}\quad \mathop{\sum }_{k=0}^{(p-1)/2}\frac{1}{2k-1}\binom{a}{k}\binom{-1-a}{k}\end{eqnarray}$$ modulo $p^{2}$ in terms of Bernoulli and Euler polynomials.

Some aspects of familiarization with binomial coefficients

2019 ◽  
pp. 49-53
Author(s):  
Abdulkarim Magomedov ◽  
S.A. Lavrenchenko
Keyword(s):  
Elementary Mathematics ◽  
Binomial Coefficients ◽  
Computational Aspects

New laconic proofs of two classical statements of combinatorics are proposed, computational aspects of binomial coefficients are considered, and examples of their application to problems of elementary mathematics are given.

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