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Menon’s identity is a classical identity involving gcd sums and the Euler totient function $φ$ . A natural generalization of $φ$ is the Klee’s function $Φ_{s}$ . We derive a Menon-type identity using Klee’s function and a generalization of the gcd function. This identity generalizes an identity given by Y. Li and D. Kim (2017).

A necessary and sufficient condition for the primality of Fermat numbers

Michal Křížek, Lawrence Somer (2001)

Mathematica Bohemica

We examine primitive roots modulo the Fermat number $F_{m} = 2^{2^{m}} + 1$ . We show that an odd integer $n \geq 3$ is a Fermat prime if and only if the set of primitive roots modulo $n$ is equal to the set of quadratic non-residues modulo $n$ . This result is extended to primitive roots modulo twice a Fermat number.

A New Condition for Consecutive Primitive Roots of a Prime.

Emanuel Vegh (1970)

Elemente der Mathematik

A new necessary condition for moduli of non-natural irreducible disjoint covering system.

Polách, I. (1994)

Acta Mathematica Universitatis Comenianae. New Series

A note on character sums over finite fields.

D.A. Burgess (1972)

Journal für die reine und angewandte Mathematik

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

Romeo Meštrović (2012)

Czechoslovak Mathematical Journal

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...

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