Development Of Number Theory Research Articles

Development of Number Theory and the Application in Cryptography

The primary focus of number theory, one of the fields of fundamental mathematics, is on the characteristics of integers. The development of cryptography also benefited greatly from the contributions of number theory. The paper primarily examines the evolution of number theory and its use in cryptography using a method of literature review. The study discovers that number theory development in the East reached a prosperous phase in the 1930s. From the 15th and 16th centuries through the 19th century, number theory flourished in the first. Number theory is still being developed, as demonstrated by the discovery of the biggest known prime number by American mathematician Curtis Cooper. Numerology is also extensively employed in cryptography, including the RSA technique and digital signatures. Numerous algorithms used in cryptography mostly depend on a working understanding of number theory.

Scientific and Practical Approaches to Improving Noise Muffler Designs of Piston Internal Combustion Engines Based on Theory of Numbers

A methodological method based on the use of the theory of preferred numbers has been proposed in order to improve the most important parameters of the working bodies of noise mufflers. As a result of many years of scientific research, the authors have established a previously unknown theoretical relationship between the main series of preferred numbers, golden ratio and Fibonacci series numbers. A new direction in the development of number theory has been considered in the paper, its classification has been compiled, including the geometric theory of numbers, preferred numbers, containing a new basic series of preferred numbers using the Fibonacci sequence. New formulas have been obtained to determine the denominators of geometric progressions for the series of preferred numbers and the area of a circle. Determining the area of a circle using the new formula allows to get more accurate values. A new formula for determining the circumference of a circle has also been derived. The designs of perforated partitions have been developed, in which the laws of the new basic series of preferred numbers are used. Determining the area of a circle using the new formula allows you to get more accurate values. A new formula for determining the circumference of a circle is also obtained. The designs of perforated partitions have been developed, in which the regularities of the new basic series of preferred numbers have been used. A calculated substantiation of the main geometric and structural dimensions of noise mufflers is given using a mathematical model of a perforated golden partition and new basic series of preferred numbers, which allow to obtain a noise muffler design that has the lowest possible aerodynamic resistance with the maximum possible reduction in the noise level of exhaust gases from internal combustion engines. An innovative model of a noise muffler for reciprocating internal combustion engines with improved hydraulic and acoustic characteristics based on the theory of numbers is proposed in the paper. The theory of preferred numbers applies to any technical device.

Arithmetic of Shimura Varieties

Arithmetic properties of Shimura varieties are an exciting topic which has roots in classical topics of algebraic geometry and of number theory such as modular curves and modular forms. This very active research field has contributed to some of the most spectacular developments in number theory and arithmetic geometry in the last twenty years. Shimura varieties and their equal characteristic analogue, moduli spaces of shtukas, are closely related to the Langlands program (classical as well as p-adic). A particular case is given by moduli spaces of abelian varieties, a classical object of study in algebraic geometry.

Some topics in recent developments in number theory

Some topics in recent developments in number theory

Relativistic causality and position space renormalization

The paper gives a historical survey of the causal position space renormalization with a special attention to the role of Raymond Stora in the development of this subject. Renormalization is reduced to subtracting the pole term in analytically regularized primitively divergent Feynman amplitudes. The identification of residues with “quantum periods” and their relation to recent developments in number theory are emphasized. We demonstrate the possibility of integration over internal vertices (that requires control over the infrared behavior) in the case of the massless φ4 theory and display the dilation and the conformal anomaly.

Starting with gaps between k-free numbers

We survey various developments in Number Theory that were inspired by classical papers by Roth [On the gaps between squarefree numbers, J. London Math. Soc. 26 (1951) 263–268] and by Halberstam and Roth [On the gaps between consecutive k-free integres, J. London Math. Soc. 26 (1951) 268–273].

JOHN KNOPFMACHER, [ABSTRACT] ANALYTIC NUMBER THEORY, AND THE THEORY OF ARITHMETICAL FUNCTIONS

Dedicated to the memory of John Knopfmacher (1937–1999) In this paper some important contributions of John Knopfmacher to ‘ Analytic Number Theory’ are described. This theory investigates semigroups with countably many generators (generalized ‘primes’), with a norm map (or a ‘degree map’), and satisfying certain conditions on the number of elements with norm less than x (Axiom A resp. Axiom A#), and ‘arithmetical’ functions defined on these semigroups. It is tried to show some of the impact of John Knopfmachers ideas to the future development of number theory, in particular for the topics ‘arithmetical functions’ and ‘asymptotics in additive arithmetical semigroups’.

Some uses of microcomputers in number theory research

The development of number theory has been influenced greatly by the use of computers. Although many of these computers were comparatively large and fast when they were used, most of the work they did could have been done by modern microcomputers. This paper describes several ways microcomputers could have been used and can be used to aid research in number theory. Since this article updates a recent survey article of H. C. Williams on the influence of computers in the development of number theory, it discusses only a few problems such as factoring integers, testing integers for primality and Waring's problem.

The influence of computers in the development of number theory

The development of number theory has been greatly influenced by the use of large scale computing devices. This paper describes several different ways in which computers have aided in the growth of various branches of this subject. Some of the topics discussed are: factoring, primality testing, the syracuse problem, Abel's problem, diophantine equations, Fermat's Last Theorem, the Twin Prime Conjecture, the Riemann Hypothesis, and some problems from algebraic number theory. A lengthy (but by no means complete) bibliography is also included.

On consecutive integer solutions for $y^{2}-k=x^{3}$

In this paper the number of k's having a given number of consecutive integer solutions either for x or for y or for both in the 2 3 equation y k = x has been found. Introduction. The equation y2 k =x3 now known as Mordell's equation, has interested mathematicians for more than three centuries, and has played an important role in the development of number theory. Many interesting, important, and beautiful results on this subject have been published. For a complete bibliography one can see [1]. It is really surprising to see that an interesting problem like finding the number of k's having a given number of solutions of a given type has been overlooked by all for such a long time. In a recent paper [2] we have given the identity ( t33) 223 (t3 + 1)2 (2t)3 = (3t3 _ 1)2 (2t2)3 = t6 6t3 + 1. For even t this shows that there is an infinite number of k's having six solutions (x, ?y) where x and y are coprime. This result also follows from the identity (4a 3 + 1)2(2a)3 = (4a3)2(-1)3 = (4a3 1)2 (2a3 = 16a6 + 1, which is obtained in this paper. We want to find the number of k's having a given number of consecutive integer solutions for y or x or both. Theorem 1. If y2 _ k = x3 (k an integer) has two consecutive integer solutions for y, then k is a positive integer. Proof. If x is a negative integer, then k = y2 _ x3 is always positive. So we may assume that x takes only positive integer values. Let (x1, y1) and (x2, y1 -1) be two integer solutions for y2 k = x3. Then we have Received by the editors November 5, 1973. AMS (MOS) subject classifications (1970). Primary lOB1O. 'This paper is dedicated to Professor M. T. Nagell, Uppsala University, Sweden. Copyright ? 1975, American Mathematical Society

On the Number of Solutions of Some General Types of Equations in a Finite Field

The conditional equation f(x1, … , xs) = 0, where f is a polynomial in the x´s with coefficients in a finite field F(pn), is connected with many well-known developments in number theory and algebra, such as: Waring's problem, the arithmetical theory of quadratic forms, the Riemann hypothesis for function fields, Fermat's Last Theorem, cyclotomy, and the theory of congruences in commutative rings.