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This document provides a summary of a special issue of the journal Mathematics, which features research papers in the field of algebraic, analytic, and computational number theory. The papers cover a wide range of topics, including generalized Lucas pseudoprimality, Fibonacci numbers, piecewise linear difference equations, properties of Euler's function, regularities in ordered semihypergroups, zero-density results for automorphic functions, properties of Fibonacci finite operator quaternions, and more. The authors of the special issue believe that the techniques used in these papers have the potential to yield new results in these areas of mathematics. [Extracted from the article]

In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field Z p for special values of prime integer p. [ABSTRACT FROM AUTHOR]

In this paper we study certain quaternion algebras and symbol algebras which split.

In this article, we find some properties of certain types of entropies of a natural number. We are studying a way of measuring the "disorder" of the divisors of a natural number. We compare two of the entropies H and H ¯ defined for a natural number. An useful property of the Shannon entropy is the additivity, H S (pq) = H S (p) + H S (q) , where pq denotes tensor product, so we focus on its study in the case of numbers and ideals. We mention that only one of the two entropy functions discussed in this paper satisfies additivity, whereas the other does not. In addition, regarding the entropy H of a natural number, we generalize this notion for ideals, and we find some of its properties. [ABSTRACT FROM AUTHOR]

In this paper we provide some applications of the norm form in some quaternion division algebras over rational field, and we give some properties of Fibonacci sequence and Fibonacci sequence in connection to quaternion elements. We provide some properties of the norm of a rational quaternion algebra, in connection to the famous Lagrange's four‐square theorem and its generalizations given by Ramanujan. Moreover, we prove some results regarding the arithmetic of integer quaternions defined on some division quaternion algebras, and we define and give properties of some special quaternions by using Fibonacci sequences. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a certain class of diagonalizable matrices, starting from a certain difference equation of degree 3, which is related to Pell's numbers. [ABSTRACT FROM AUTHOR]

The article is a review of the book "Diophantine m-tuples and Elliptic Curves" by Andrej Dujella. The book focuses on the specialized topic of Diophantine m-tuples in number theory, which are related to elliptic curves and certain Diophantine equations. The book contains five chapters, covering topics such as the theory of elliptic curves, methods for calculating the rank of an elliptic curve, and finding integer points on elliptic curves. The reviewer praises the book as a comprehensive and valuable resource for current research in number theory. [Extracted from the article]

In this paper, we characterize residuated lattices for which the topological space of prime ideals is a Noetherian space. The notion of i-Noetherian residuated lattice is introduced and related properties are investigated. We proved that a residuated lattice is i-Noetherian iff every ideal is principal. Moreover, we show that a residuated lattice has the spectrum of a Noetherian space iff it is i-Noetherian. [ABSTRACT FROM AUTHOR]

In this paper, we provide properties and applications of some special integer sequences. We generalize and give some properties of Pisano period. Moreover, we provide a new application in cryptography and applications of some quaternion elements. [ABSTRACT FROM AUTHOR]

Sunt date aplicaţii ale următoarelor identităţi combinatoriale: 1. ∑k=0p Cmk . Cnp-k = Cm+np, 2. ∑k=rn Cnk . Ckr = Cnr . 2n-r, ∑k=0n Cm+kk . 2n-k + ∑k=0m Cn+kk . 2m-k = 2m+n+1, identitatea lui Banach. [ABSTRACT FROM AUTHOR]

Symbol algebras have many applications in various domains of mathe- matics, as for example number theory (class field theory). Since these algebras are a natural generalization of the quaternion algebras, in this short survey we present some properties of the quaternion algebras and of the symbol algebras of prime degree. [ABSTRACT FROM AUTHOR]

We give a characterization of the prime natural numbers p of the form xq + ryq, where q and r are distinct odd primes. [ABSTRACT FROM AUTHOR]

We give a characterization of the prime natural numbers p of the form xq + ryq, where q and r are distinct odd primes. [ABSTRACT FROM AUTHOR]

In this paper, we find some inequalities which involve Euler's function, extended Euler's function, the function τ , and the generalized function τ in algebraic number fields. [ABSTRACT FROM AUTHOR]

Let (t n (r)) n ≥ 0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence t n (r) = t n − 1 (r) + ⋯ + t n − r (r) for n ≥ r , with initial values t 0 (r) = 0 and t i (r) = 1 , for all 1 ≤ i ≤ r . In 2002, Grossman and Luca searched for terms of the sequence (t n (2)) n , which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ℓ ≥ 1 , there exists an effectively computable constant C = C (ℓ) > 0 (only depending on ℓ), such that, if (m , n , r) is a solution of t m (r) = n ! + (n + 1) ! + ⋯ + (n + ℓ) ! , with r even, then max { m , n , r } < C . As an application, we solve the previous equation for all 1 ≤ ℓ ≤ 5 . [ABSTRACT FROM AUTHOR]

We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some recent arithmetic properties of the generalized Lucas, and generalized Pell–Lucas sequences, to define some new types of pseudoprimes of levels k + and k − and parameter a. For these novel pseudoprime sequences we investigate some basic properties and calculate numerous associated integer sequences which we have added to the Online Encyclopedia of Integer Sequences. [ABSTRACT FROM AUTHOR]

In this paper, we find some properties of Euler's function and Dedekind's function. We also generalize these results, from an algebraic point of view, for extended Euler's function and extended Dedekind's function, in algebraic number fields. Additionally, some known inequalities involving Euler's function and Dedekind's function, we generalize them for extended Euler's function and extended Dedekind's function, working in a ring of integers of algebraic number fields. [ABSTRACT FROM AUTHOR]

The article presents the book chapters of the book ""Why Quadratic Diophantine Equations?", by Titu Andreescu. It mentions that first chapter focuses on the Thue's theorem. The second chapter of the book called "Continued Fractions, Diophantine Approximation, and Quadratic Rings", depicts the theory of the continued fractions.