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### 17 necessary and sufficient conditions for the primality of Fermat numbers

Acta Mathematica et Informatica Universitatis Ostraviensis

### A generalization of a necessary and sufficient condition for primality due to Vantieghem.

International Journal of Mathematics and Mathematical Sciences

Acta Arithmetica

### A necessary and sufficient condition for the primality of Fermat numbers

Mathematica Bohemica

We examine primitive roots modulo the Fermat number ${F}_{m}={2}^{{2}^{m}}+1$. We show that an odd integer $n\ge 3$ 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 Fibonacci-like sequence of composite numbers.

Journal of Integer Sequences [electronic only]

### A note on factorization of the Fermat numbers and their factors of the form $3h{2}^{n}+1$

Mathematica Bohemica

We show that any factorization of any composite Fermat number ${F}_{m}={{2}^{2}}^{m}+1$ into two nontrivial factors can be expressed in the form ${F}_{m}=\left(k{2}^{n}+1\right)\left(\ell {2}^{n}+1\right)$ for some odd $k$ and $\ell ,k\ge 3,\ell \ge 3$, and integer $n\ge m+2,3n<{2}^{m}$. We prove that the greatest common divisor of $k$ and $\ell$ is 1, $k+\ell \equiv 0\phantom{\rule{4pt}{0ex}}mod{2}^{n},\phantom{\rule{4pt}{0ex}}max\left(k,\ell \right)\ge {F}_{m-2}$, and either $3|k$ or $3|\ell$, i.e., $3h{2}^{m+2}+1|{F}_{m}$ for an integer $h\ge 1$. Factorizations of ${F}_{m}$ into more than two factors are investigated as well. In particular, we prove that if ${F}_{m}={\left(k{2}^{n}+1\right)}^{2}\left(\ell {2}^{j}+1\right)$ then $j=n+1,3|\ell$ and $5|\ell$.

### A note on pseudoprimes with respect to abelian linear recurring sequence

Mathematica Slovaca

### A sharp estimate of the number of integral points in a 4-dimensional tetrahedra.

Journal für die reine und angewandte Mathematik

### Algebraic and combinatorial properties of products [Abstract of thesis]

Commentationes Mathematicae Universitatis Carolinae

### Appending digits to generate an infinite sequence of composite numbers.

Journal of Integer Sequences [electronic only]

### Bounds for frequencies of residues of second-order recurrences modulo ${p}^{r}$

Mathematica Bohemica

The authors examine the frequency distribution of second-order recurrence sequences that are not $p$-regular, for an odd prime $p$, and apply their results to compute bounds for the frequencies of $p$-singular elements of $p$-regular second-order recurrences modulo powers of the prime $p$. The authors’ results have application to the $p$-stability of second-order recurrence sequences.

### Bounds for the counting function of the Jordan-Pólya numbers

Archivum Mathematicum

A positive integer $n$ is said to be a Jordan-Pólya number if it can be written as a product of factorials. We obtain non-trivial lower and upper bounds for the number of Jordan-Pólya numbers not exceeding a given number $x$.

### Categories of Semigroups with Divisor Theory

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

### Characterization of power digraphs modulo $n$

Commentationes Mathematicae Universitatis Carolinae

A power digraph modulo $n$, denoted by $G\left(n,k\right)$, is a directed graph with ${Z}_{n}=\left\{0,1,\cdots ,n-1\right\}$ as the set of vertices and $E=\left\{\left(a,b\right):{a}^{k}\equiv b\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}n\right)\right\}$ as the edge set, where $n$ and $k$ are any positive integers. In this paper we find necessary and sufficient conditions on $n$ and $k$ such that the digraph $G\left(n,k\right)$ has at least one isolated fixed point. We also establish necessary and sufficient conditions on $n$ and $k$ such that the digraph $G\left(n,k\right)$ contains exactly two components. The primality of Fermat number is also discussed.

### Cipolla pseudoprimes.

Journal of Integer Sequences [electronic only]

### Combinatorial properties of products of graphs

Czechoslovak Mathematical Journal

Acta Arithmetica

### Correction to the paper: On the sum-of-Divisors Function of the numbers of Fermat and Ferentinou-Nicolacopoulou

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

### Cribrum algebraicum oder die cofunctionale Entstehung der Primzahlen.

Jahresbericht der Deutschen Mathematiker-Vereinigung

### Die Primfaktorzerlegung der Werte der Kreisteilungspolynome.

Journal für die reine und angewandte Mathematik

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