1. On the asymptotic density of fermat pseudoprimes to a fixed baseSlobodan Filipovski, 2026, izvirni znanstveni članek Opis: Fermat's little theorem provides a classical and widely used test for primality: if $n$ is prime and $\gcd(a,n)=1$, then \[ a^{n-1}\equiv 1 \pmod n. \] Composite integers satisfying this congruence are called Fermat pseudoprimes to base $a$, and they represent a fundamental obstruction to Fermat-type primality tests. A classical result of Erd\H{o}s shows that, for each fixed base $a\geq 2$, the set of such numbers has asymptotic density zero. In this note we give a short and elementary proof of this density-zero property for Fermat pseudoprimes, based on ideas of Erd\H{o}s and Niven.
Ključne besede: Fermat pseudoprimes, asymptotic density, primality testing
Objavljeno v RUP: 21.09.2026; Ogledov: 285; Prenosov: 13
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2. Some results on ▫$\sigma_t$▫-irregularitySlobodan Filipovski, Darko Dimitrov, Martin Knor, Riste Škrekovski, 2026, izvirni znanstveni članek Opis: The (\sigma_t)-irregularity (or sigma total index) is a graph invariant defined as [ \sigma_t(G)=\sum_{{u,v}\subseteq V(G)}(d(u)-d(v))^2, ] where (d(z)) denotes the degree of a vertex (z). This irregularity measure was proposed by Réti in 2019 and recently rediscovered by Dimitrov and Stevanović in 2023. In this paper, we remark that (\sigma_t(G)=n^2\operatorname{Var}(G)), where (\operatorname{Var}(G)) is the degree variance of the graph. We show that among all complete bipartite graphs on (n) vertices, one of the corresponding complete bipartite graphs whose part sizes are closest to (n(2-\sqrt{2})/4) and (n(2+\sqrt{2})/4) has the maximum sigma total index. Moreover, various upper and lower bounds for (\sigma_t)-irregularity are provided. In this direction, we establish a relation between the graph energy (\mathcal{E}(G)) and (\sigma_t)-irregularity and derive bounds related to the Laplacian eigenvalues of the graph.
Ključne besede: irregularity, total irregularity, energy of graphs, Laplacian eigenvalues
Objavljeno v RUP: 14.08.2026; Ogledov: 422; Prenosov: 7
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4. Exploring the bounds and relationships of chemical graph indices : master's thesisArbër Avdullahu, 2023, magistrsko delo Ključne besede: energy of a graph, Randić index, first Zagreb index, IRB index, extremal graph, spectrum, eigenvalue, metaheuristic
Objavljeno v RUP: 05.10.2023; Ogledov: 2612; Prenosov: 77
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