Finite Arithmetic Axiomatization for the Basis of Hyperrational Non-Standard Analysis

The standard elementary number theory is not a finite axiomatic system due to the presence of the induction axiom scheme. Absence of a finite axiomatic system is not an obstacle for most tasks, but may be considered as imperfect since the induction is strongly associated with the presence of set theory external to the axiomatic system. Also in the case of logic approach to the artificial intelligence problems presence of a finite number of basic axioms and states is important. Axiomatic hyperrational analysis is the axiomatic system of hyperrational number field. The properties of hyperrational numbers and functions allow them to be used to model real numbers and functions of classical elementary mathematical analysis. However hyperrational analysis is based on well-known non-finite hyperarithmetic axiomatics. In the article we present a new finite first-order arithmetic theory designed to be the basis of the axiomatic hyperrational analysis and, as a consequence, mathematical analysis in general as a basis for all mathematical application including AI problems. It is shown that this axiomatics meet the requirements, i.e., it could be used as the basis of an axiomatic hyperrational analysis. The article in effect completes the foundation of axiomatic hyperrational analysis without calling in an arithmetic extension, since in the framework of the presented theory infinite numbers arise without invoking any new constants. The proposed system describes a class of numbers in which infinite numbers exist as natural objects of the theory itself. We also do not appeal to any “enveloping” set theory.

A note on the polynomial-exponential Diophantine equation (a^n − 1)(b^n − 1) = x^2

For any positive integer t, let ord_2 t denote the order of 2 in the factorization of t. Let a,\,b be two distinct fixed positive integers with \min\{a,b\}>1. In this paper, using some elementary number theory methods, the existence of positive integer solutions (x,n) of the polynomial-exponential Diophantine equation (*) (a^n-1)(b^n-1)=x^2 with n>2 is discussed. We prove that if \{a,b\}\ne \{13,239\} and ord_2(a^2-1)\ne ord_2(b^2-1), then (*) has no solutions (x,n) with 2\mid n. Thus it can be seen that if \{a,b\}\equiv \{3,7\},\{3,15\},\{7,11\},\{7,15\} or \{11,15\} \pmod{16}, where \{a,b\} \equiv \{a_0,b_0\} \pmod{16} means either a \equiv a_0 \pmod{16} and b \equiv b_0\pmod{16} or a\equiv b_0 \pmod{16} and b\equiv a_0 \pmod{16}, then (*) has no solutions (x,n).

SOLUTIONS TO A LEBESGUE–NAGELL EQUATION

We find all integer solutions to the equation $x^2+5^a\cdot 13^b\cdot 17^c=y^n$ with $a,\,b,\,c\geq 0$ , $n\geq 3$ , $x,\,y>0$ and $\gcd (x,\,y)=1$ . Our proof uses a deep result about primitive divisors of Lucas sequences in combination with elementary number theory and computer search.

Elementary Number Theory Problems. Part II

Summary In this paper problems 14, 15, 29, 30, 34, 78, 83, 97, and 116 from [6] are formalized, using the Mizar formalism [1], [2], [3]. Some properties related to the divisibility of prime numbers were proved. It has been shown that the equation of the form p 2 + 1 = q 2 + r 2, where p, q, r are prime numbers, has at least four solutions and it has been proved that at least five primes can be represented as the sum of two fourth powers of integers. We also proved that for at least one positive integer, the sum of the fourth powers of this number and its successor is a composite number. And finally, it has been shown that there are infinitely many odd numbers k greater than zero such that all numbers of the form 22 n + k (n = 1, 2, . . . ) are composite.

Lorenzen’s Reshaping of Krull’s Fundamentalsatz for Integral Domains (1938–1953)

Krull’s Fundamentalsatz, the generalisation of the main theorem of elementary number theory to integral domains, is the starting point of Lorenzen’s career in mathematics. This article traces a conceptual history of Lorenzen’s successive reformulations of the Fundamentalsatz on the basis of excerpts of his articles. An edition of the extant correspondence of Lorenzen with Hasse, Krull, and Aubert provides a better understanding of the context of these investigations.

Control of connectivity and rigidity in prismatic assemblies

How can we manipulate the topological connectivity of a three-dimensional prismatic assembly to control the number of internal degrees of freedom and the number of connected components in it? To answer this question in a deterministic setting, we use ideas from elementary number theory to provide a hierarchical deterministic protocol for the control of rigidity and connectivity. We then show that it is possible to also use a stochastic protocol to achieve the same results via a percolation transition. Together, these approaches provide scale-independent algorithms for the cutting or gluing of three-dimensional prismatic assemblies to control their overall connectivity and rigidity.

3n+1 Problem and its Dynamics

The subject of this paper is the well-known 3n + 1 problem of elementary number theory. This problem concerns with the behaviour of the iteration of a function which takes odd integers n to 3n + 1, and even integers n to n/2. There is a famous Collatz conjecture associated to this problem which asserts that, starting from any positive integer n, repeated iteration of the function eventually produces the value. We briefly discuss some basic facts and results of 3n + 1 problem and Collatz conjecture. Basically, we more concentrate on the generalization of this problem and conjecture to holomorphic dynamics.