| Title:
|
A note on the congruence ${np^k\choose mp^k} \equiv {n\choose m} \pmod {p^r}$ (English) |
| Author:
|
Meštrović, Romeo |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
62 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
59-65 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
congruence |
| Keyword:
|
prime powers |
| Keyword:
|
Lucas' theorem |
| Keyword:
|
Wolstenholme prime |
| Keyword:
|
set $W(k,r)$ |
| MSC:
|
11A07 |
| MSC:
|
11B65 |
| idZBL:
|
Zbl 1249.11031 |
| idMR:
|
MR2899734 |
| DOI:
|
10.1007/s10587-012-0016-7 |
| . |
| Date available:
|
2012-03-05T07:11:08Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142040 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
|
[8] Meštrović, R.: A note on the congruence ${nd\choose md}\equiv{n\choose m}\pmod{q}$.Am. Math. Mon. 116 (2009), 75-77. MR 2478756 |
| Reference:
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| Reference:
|
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