# A note on the congruence $\left(\genfrac{}{}{0pt}{}{n{p}^{k}}{m{p}^{k}}\right)\equiv \left(\genfrac{}{}{0pt}{}{n}{m}\right)\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}{p}^{r}\right)$

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

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## Abstract

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In the paper we discuss the following type congruences: $\left(\genfrac{}{}{0pt}{}{n{p}^{k}}{m{p}^{k}}\right)\equiv \left(\genfrac{}{}{0pt}{}{m}{n}\right)\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}{p}^{r}\right),$ 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 $W\left(k,r\right)$ 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 $W\left(k,r\right)$ and inclusion relations between them for various values $k$ and $r$. In particular, we prove that $W\left(k+i,r\right)=W\left(k-1,r\right)$ for all $k\ge 2$, $i\ge 0$ and $3\le r\le 3k$, and $W\left(k,r\right)=W\left(1,r\right)$ 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.

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