A note on the congruence ${np^k\atopwithdelims ()mp^k} \equiv {n\atopwithdelims ()m} \hspace{4.44443pt}(\@mod \; p^r)$

Romeo Meštrović

A note on the congruence $(\binom{n p^{k}}{m p^{k}}) \equiv (\binom{n}{m}) (mod p^{r})$

Romeo Meštrović

Czechoslovak Mathematical Journal (2012)

Volume: 62, Issue: 1, page 59-65
ISSN: 0011-4642

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In the paper we discuss the following type congruences:

(\binom{n p^{k}}{m p^{k}}) \equiv (\binom{m}{n}) (mod p^{r}),

where

p

is a prime,

n

,

m

,

k

and

r

are various positive integers with

n \geq m \geq 1

,

k \geq 1

and

r \geq 1

. Given positive integers

k

and

r

, denote by

W (k, r)

the set of all primes

p

such that the above congruence holds for every pair of integers

n \geq m \geq 1

. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets

W (k, r)

and inclusion relations between them for various values

k

and

r

. In particular, we prove that

W (k + i, r) = W (k - 1, r)

for all

k \geq 2

,

i \geq 0

and

3 \leq r \leq 3 k

, and

W (k, r) = W (1, r)

for all

3 \leq r \leq 6

and

k \geq 2

. We also noticed that some of these properties may be used for computational purposes related to congruences given above.

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Meštrović, Romeo. "A note on the congruence ${np^k\atopwithdelims ()mp^k} \equiv {n\atopwithdelims ()m} \hspace{4.44443pt}(\@mod \; p^r)$." Czechoslovak Mathematical Journal 62.1 (2012): 59-65. <http://eudml.org/doc/246182>.

@article{Meštrović2012,
abstract = {In the paper we discuss the following type congruences: \[ \biggl (\{np^k\atop mp^k\}\biggr ) \equiv \left(m \atop n\right) \hspace\{10.0pt\}(\@mod \; p^r), \] where $p$ is a prime, $n$, $m$, $k$ and $r$ are various positive integers with $n\ge m\ge 1$, $k\ge 1$ and $r\ge 1$. Given positive integers $k$ and $r$, denote by $W(k,r)$ the set of all primes $p$ such that the above congruence holds for every pair of integers $n\ge m\ge 1$. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets $W(k,r)$ and inclusion relations between them for various values $k$ and $r$. In particular, we prove that $W(k+i,r)=W(k-1,r)$ for all $k\ge 2$, $i\ge 0$ and $3\le r\le 3k$, and $W(k,r)=W(1,r)$ for all $3\le r\le 6$ and $k\ge 2$. We also noticed that some of these properties may be used for computational purposes related to congruences given above.},
author = {Meštrović, Romeo},
journal = {Czechoslovak Mathematical Journal},
keywords = {congruence; prime powers; Lucas’ theorem; Wolstenholme prime; set $W(k,r)$; congruence; prime powers; Lucas' theorem; Wolstenholme prime; set },
language = {eng},
number = {1},
pages = {59-65},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A note on the congruence $\{np^k\atopwithdelims ()mp^k\} \equiv \{n\atopwithdelims ()m\} \hspace\{4.44443pt\}(\@mod \; p^r)$},
url = {http://eudml.org/doc/246182},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Meštrović, Romeo
TI - A note on the congruence ${np^k\atopwithdelims ()mp^k} \equiv {n\atopwithdelims ()m} \hspace{4.44443pt}(\@mod \; p^r)$
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 59
EP - 65
AB - In the paper we discuss the following type congruences: \[ \biggl ({np^k\atop mp^k}\biggr ) \equiv \left(m \atop n\right) \hspace{10.0pt}(\@mod \; p^r), \] where $p$ is a prime, $n$, $m$, $k$ and $r$ are various positive integers with $n\ge m\ge 1$, $k\ge 1$ and $r\ge 1$. Given positive integers $k$ and $r$, denote by $W(k,r)$ the set of all primes $p$ such that the above congruence holds for every pair of integers $n\ge m\ge 1$. Using Ljunggren’s and Jacobsthal’s type congruences, we establish several characterizations of sets $W(k,r)$ and inclusion relations between them for various values $k$ and $r$. In particular, we prove that $W(k+i,r)=W(k-1,r)$ for all $k\ge 2$, $i\ge 0$ and $3\le r\le 3k$, and $W(k,r)=W(1,r)$ for all $3\le r\le 6$ and $k\ge 2$. We also noticed that some of these properties may be used for computational purposes related to congruences given above.
LA - eng
KW - congruence; prime powers; Lucas’ theorem; Wolstenholme prime; set $W(k,r)$; congruence; prime powers; Lucas' theorem; Wolstenholme prime; set
UR - http://eudml.org/doc/246182
ER -

References

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Meštrović, R., A note on the congruence $(\binom{n d}{m d}) \equiv (\binom{n}{m}) (mod q)$ , Am. Math. Mon. 116 (2009), 75-77. (2009) MR2478756
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