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We construct a new arithmetic-term representation for the function gcd(a, b). As a byproduct, we also deduce a representation gcd(a, b) by a modular term in integer arithmetic., Comment: 5 pages

We present closed forms for several functions that are fundamental in number theory and we explain the method used to obtain them. Concretely, we find formulas for the p-adic valuation, the number-of-divisors function, the sum-of-divisors function, Euler's totient function, the modular inverse, the integer part of the root, the integer part of the logarithm, the multiplicative order and the discrete logarithm. Although these are very complicated, they only involve elementary operations, and to our knowledge no other closed form of this kind is known for the aforementioned functions.

An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are integers. While in the paper "On the representation of C-recursive integer sequences by arithmetic terms" by Prunescu and Sauras-Altuzarra, the terms consist of the remainder operation, applied on a division; the representations shown here are a division applied to a remainder operation, respectively the composition of two remainder operations., Comment: 22 pages

We show that, if an integer sequence is given by a linear recurrence of constant rational coefficients, then it can be represented as the difference of two arithmetic terms with exponentiation, which do not contain any irrational constant. We apply our methods to various Lucas sequences including the classical Fibonacci sequence, to the sequence of solutions of the Pell equation and to some natural C-recursive sequences of degree 3., Comment: 20 pages

The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of algebraic relations found by the algorithm consists of polynomials having coefficients in the prime field F_p., Comment: In Proceedings FROM 2023, arXiv:2309.12959

Linear logic is a substructural logic proposed as a refinement of classical and intuitionistic logics, with applications in programming languages, game semantics, and quantum physics. We present a template for Gentzen-style linear logic sequents that supports verification of logic inference rules using automatic theorem proving. Specifically, we use the Z3 Theorem Prover [8] to check targeted inference rules based on a set of inference rules that are presumed to be valid. To demonstrate the approach, we apply it to validate several derived inference rules for two different fragments of linear logic: MLL+Mix (Multiplicative Linear Logic extended with a Mix rule) and MILL (Multiplicative Intuitionistic Linear Logic)., Comment: In Proceedings FROM 2023, arXiv:2309.12959

It has been said that Maxwell's theory of electromagnetic field is relativistic as Einstein showed that these axioms of Maxwell are all Lorentz invariant. We investigate some issues regarding these results., Comment: 23 pages, Physics Beyond Relativity conference, October 20, 2019, Prague, Czech Republic

We review the special theory of relativity kinematics and dynamics as well as the general theory of relativity under the aspect of logic., Comment: 28 pages, Physics Beyond Relativity conference, October 20, 2019, Prague, Czech Republic

We present an evolutionary self-governing model based on the numerical atomic rule $Z(a,b)=ab/\gcd(a,b)^2$, for $a,b$ positive integers. Starting with a sequence of numbers, the initial generation $Gin$, a new sequence is obtained by applying the $Z$-rule to any neighbor terms. Likewise, applying repeatedly the same procedure to the newest generation, an entire matrix $T_{Gin}$ is generated. Most often, this matrix, which is the recorder of the whole process, shows a fractal aspect and has intriguing properties. If $Gin$ is the sequence of positive integers, in the associated matrix remarkable are the distinguished geometrical figures called the $Z$-solitons and the sinuous evolution of the size of numbers on the western edge. We observe that $T_{\mathbb{N}^*}$ is close to the analogue free of solitons matrix generated from an initial generation in which each natural number is replaced by its largest divisor that is a product of distinct primes. We describe the shape and the properties of this new matrix. N. J. A. Sloane raised a few interesting problems regarding the western edge of the matrix $T_{\mathbb{N}^*}$. We solve one of them and present arguments for a precise conjecture on another., Comment: 20 pages, 9 figures

A primitive type of two-dimensional dynamic system is introduced. It is shown that there is no decision procedure able to answer if such a dynamical system is ultimately zero., Comment: 7 pages

Some linear dynamical systems over finite fields are studied and the self-similar character of their development is proved. Connections with aperiodic tilings, Delanoy numbers and other topics are also proved. The prime fields F_p have a canonical presentation as sets of self-similar carpets. The multiplicative inverse corresponds to mirroring., Comment: 14 pages, some latex figures in text (no separate files) These results have been presented at CiE 2007, Siena, Italy