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In the paper we discuss the following type congruences: $$\left(\genfrac{}{}{0pt}{}{n{p}^k}{m{p}^k}\right)\equiv \left(\genfrac{}{}{0pt}{}{m}{n}\right)(mod{p}^r),$$ 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...

Let $p>3$ be a prime, and let $q_p\left(2\right)=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base $2$. In this note we prove that $$\sum _{k=1}^{p-1}\frac{1}{k·2^k}\equiv q_p\left(2\right)-\frac{p{q}_p{\left(2\right)}^2}{2}+\frac{p^2{q}_p{\left(2\right)}^3}{3}-\frac{7}{48}{p}^2{B}_{p-3}(mod{p}^3),$$ which is a generalization of a congruence due to Z. H. Sun. Our proof is based on certain combinatorial identities and congruences for some alternating harmonic sums. Combining the above congruence with two congruences by Z. H. Sun, we show that $$q_p{\left(2\right)}^3\equiv -3\sum _{k=1}^{p-1}\frac{2^k}{k^3}+\frac{7}{16}\sum _{k=1}^{(p-1)/2}\frac{1}{k^3}(modp),$$ which is just a result established by K. Dilcher and L. Skula. As another application, we obtain a congruence for the sum ${\sum }_{k=1}^{p-1}1/(k^2·2^k)$ modulo $p^2$ that also generalizes a...

A prime $p$ is said to be a Wolstenholme prime if it satisfies the congruence $\left(\genfrac{}{}{0pt}{}{2p-1}{p-1}\right)\equiv 1(mod{p}^4)$. For such a prime $p$, we establish an expression for $\left(\genfrac{}{}{0pt}{}{2p-1}{p-1}\right)(mod{p}^8)$ given in terms of the sums $R_i:={\sum }_{k=1}^{p-1}1/k^i$ ($i=1,2,3,4,5,6)$. Further, the expression in this congruence is reduced in terms of the sums $R_i$ ($i=1,3,4,5$). Using this congruence, we prove that for any Wolstenholme prime $p$ we have $$\left(\genfrac{}{}{0pt}{}{2p-1}{p-1}\right)\equiv 1-2p\sum _{k=1}^{p-1}\frac{1}{k}-2p^2\sum _{k=1}^{p-1}\frac{1}{k^2}(mod{p}^7).$$ Moreover, using a recent result of the author, we prove that a prime $p$ satisfying the above congruence must necessarily be a Wolstenholme prime. Furthermore, applying a technique...

Let $p\ge 5$ be a prime, and let $q_p\left(2\right):=(2^{p-1}-1)/p$ be the Fermat quotient of $p$ to base $2$. The following curious congruence was conjectured by L. Skula and proved by A. Granville $$q_p{\left(2\right)}^2\equiv -\sum _{k=1}^{p-1}\frac{2^k}{k^2}(modp).$$ In this note we establish the above congruence by entirely elementary number theory arguments.