Fundamental Perceptions in Contemporary Number Theory

British Journal of Mathematics & Computer Science, 2015

The main purpose of the present article is to give a brief description of the "golden" number theory and new properties of natural numbers following from it, in particular, Z-property, D-property, Φ-code, Fcode, L-code. These properties are of big theoretical interest for number theory and can be used in computer science. The article is written in popular form and is intended for a wide circle of mathematicians (including mathematics students) and specialists in computer science, who are interested in the histories of mathematics and new ideas in the development of number theory and its applications in computer science.

1998

Introduction The general structure of this work reeects the main problems that I studied as a PhD student at the University of Pisa (1993{1997). There are ve chapters, which deal with three diierent topics: Practical numbers (Chapters 1 and 2); Sum-free sequences (Chapter 3); Arithmetical identities related to the theory of modular forms (Chapters 4 and 5). In Chapter 1 we extensively survey the theory of practical numbers, i.e., those positive integers m such that every positive integer n < m can be represented as a sum of distinct positive divisors of m. This theory has recently received attention for some unexpected similarities with the properties of primes. We deal with both arithmetical and analytical aspects of the theory. Among other things, we prove the analogue of Goldbach's conjecture for practical numbers, showing that every even positive integer can be expressed as a sum of two practical numbers. This result gives a positive answer to a conjecture raised in 1984 ...

In this paper we collected problems, which was either proposed or follow directly from results in our papers.

Dr. M. Shanmuga Sundari* Dr. S. Sagathevan**

Current study explains the concept of Algebraic Number Theory and its applications. Study was based on the literature and descriptive in nature. Algebraic number theory is a rich and diverse subfield of abstract algebra and number theory, applying the concepts of number fields and algebraic numbers to number theory to improve upon applications such as prime factorization and primarily testing. In this study, researchers begin with an overview of algebraic number fields and algebraic numbers and then move into some important results of algebraic number theory, focusing on the quadratic, or Gauss reciprocity law.

2012

In this paper, we will discuss two questions, the primality test for Fermat number Fn and several fundamental problems in transfinite number theory. For the primality test for Fermat number Fn, this paper introduces the method of primality test for Fermat number Fn and the Fermat factorization theorem. Meanwhile, the factorization theorem is used to write operation procedure, so that we can distinguish whether 2n+2t+1 is the divisor of Fn or not. For some of the fundamental problems in transfinite number theory, this paper introduces the concept of countable expansion, and then inspects countability of the set of real numbers from the aspect of expansion. Besides, we analyze the cardinal number of linear point set, deriving the fact that their cardinal number possesses invariance.

In this booklet, I present my proofs of open conjectures on the theory of numbers. It concerns the following conjectures: - The Riemann Hypothesis. - Beal's conjecture. - The conjecture $c<rad^{1.63}(abc)$. - The explicit $abc$ conjecture of Alan Baker. - Two proofs of the $abc$ conjecture. - The conjecture $c<rad^2(abc)$.

A New Theory of Numbers, 2020

“How can all of this be true all at the same time?” This will be the question you will be asking yourself once you discover the amazing inner world hiding behind numbers, as they reveal palindromes, two types of dual characteristics, visible and invisible patterns, perfect plus/minus as well as odd/even balances, and much more. Learn that prime numbers can be organized in a perfect 24-based, yet decimally-based system and aren’t randomly distributed. Discover a whole new way to see numbers as one unified, and I dare say, Intelligent and Logical system. An entire new Number Theory is hereby introduced as well. Find out further aspects hiding behind the Fibonacci numbers, and similarly found ratios point to the square roots of whole numbers. A bonus chapter reveals the number present in our solar system.

2019

We include here some corrections with respect to the published version. 1. We shall make use of this extra information later on. 2. See [29] for a short proof.