A note on the congruence ${np^k\choose mp^k} \equiv {n\choose m} \pmod {p^r}$

Title:

A note on the congruence ${np^k\choose mp^k} \equiv {n\choose m} \pmod {p^r}$

Creator:

Meštrović, Romeo

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:6eb0f786-1130-42d8-8e40-b62ab6518b66
uuid:6eb0f786-1130-42d8-8e40-b62ab6518b66

Subject:

congruence, prime powers, Lucas' theorem, Wolstenholme prime, and set $W(k,r)$

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

In the paper we discuss the following type congruences: $$ \biggl ({np^k\atop mp^k}\biggr ) \equiv \left (m \atop n\right ) \pmod {p^r}, $$ 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.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

59-65

Source:

Czechoslovak Mathematical Journal | 2012 Volume:62 | Number:1

Harvested from:

CDK

Metadata only:

false