scholarly journals Про Эйлера и Чебышёва, основателей российской математики

2021 ◽  
pp. 42-63
Author(s):  
A. Papadopoulos
Keyword(s):  
Number Theory ◽  
Continued Fractions ◽  
Applied Science ◽  
Pure Mathematics ◽  
Mathematics Integration

On the occasion of Chebyshev’s twohundredth anniversary, I review part of his work, showing that in several respects he was the heir of Euler. In doing this, I consider the works of Euler and Chebyshev on three topics in applied science: industrial machines, ballistics and geography, and then on three topics in pure mathematics: integration, continued fractions and number theory, showing that in each filed the two mathematicians were interested in the same kind of questions. По случаю двухсотлетнего юбилея Чебышёва я изучил часть его работ и пришел к выводу, что в целом ряде моментов он продолжает научные традиции Эйлера. Я рассматривал работы Эйлера и Чебышёва по трем прикладным темам: промышленное оборудование, баллистика и география, а также по трем разделам чистой математики: интегральное исчисление, непрерывные дроби и теория чисел. Показано, что во всех случаях оба математика изучали один и тот же круг вопросов.

Catching Proteus: the collaborations of Wallis and Brounker. II. Number problems

2000 ◽  
Vol 54 (3) ◽  
pp. 317-331 ◽  
Author(s):  
Jacqueline A. Stedall
Keyword(s):  
Number Theory ◽  
Continued Fractions

Following the discussion of Brouncker's work on quadrature, rectification and continued fractions in Part I of this paper, Part II analyses the disputes between Brouncker and Wallis in England and Fermat in France over problems in what would now be called number theory. Contemporary and later observers regarded Brouncker and Wallis as equally responsible for the English contribution, but it was Brouncker who took the greater interest in Fermat's challenges and who produced the more sophisticated solutions. The paper ends with an assessment of Brouncker's contribution to mathematics and argues that his contemporary reputation was well deserved and his mathematics of lasting value.

scholarly journals On Generalized Solutions of Linear Congruence ax ≡ b (mod n) for Large Modulus n

2018 ◽  
Vol 31 (1) ◽  
Author(s):  
Polemer M. Cuarto
Keyword(s):  
Number Theory ◽  
Research Method ◽  
Generalized Solution ◽  
Generalized Solutions ◽  
Linear Congruence ◽  
Public Key Cryptosystems ◽  
Alternative Method ◽  
Pure Mathematics ◽  
Cryptographic Algorithms ◽  
The Given

Number Theory, a branch of Pure Mathematics, is crucial in cryptographic algorithms. Many cryptographic systems depend heavily on some topics of Number Theory. One of these topics is the linear congruence. In cryptography, the concept of linear congruence is used to directly underpin public key cryptosystems during the process of ciphering and deciphering codes. Thus, linear congruence plays a very important role in cryptography. This paper aims to develop an alternative method and generalized solutions for solving linear congruence ax ≡ b (mod n). This study utilized expository-developmental research method. As a result, the alternative method considered two cases: (1) when (a,n) = 1 and  (2) when (a,n) > 1. The basic idea of the method is to convert the given congruence ax ≡ b (mod n) to ax = b + kn for some k, reduce modulus n by interchanging a and n, simplify the new congruence and perform the process recursively until obtaining a congruence that is trivial to solve. The advantage of this method over the existing approaches is that it can solve congruence even for large modulus n with much more efficiency. Generalized solution of linear congruence ax ≡ b (mod n) considering both cases was obtained in this study.

scholarly journals The Complexity of Mathematics

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Number Theory ◽  
Complexity Theory ◽  
Riemann Hypothesis ◽  
Open Problems ◽  
Mathematical Proofs ◽  
Pure Mathematics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

scholarly journals A theorem on numbers of the form 10^x

2019 ◽  
Author(s):  
Ravin Kumar
Keyword(s):  
Number Theory ◽  
Computer Science ◽  
Natural Numbers ◽  
The Core ◽  
Pure Mathematics ◽  
Major Application

Number theory is one of the core branches of pure mathematics. It has played an important role in the study of natural numbers. In this paper, we are presenting a theorem on the numbers of form 10^x , where x ∊ Z+ . The proposed theorem have a major application in computer science. It can be used to predict ‘n’ bits which will always represent more than 10^x total numbers. We proved that the nature of the ‘n’ bits is always one of the forms 10i, 10i + 4, or 10i + 7, where i ∊ Z+ .

scholarly journals Fundamental Theorems of Pure Mathematics & Absolute Geometry

2021 ◽  
Vol 4 (2) ◽  
Keyword(s):  
Field Theory ◽  
Number Theory ◽  
Fundamental Theorem ◽  
Absolute Geometry ◽  
Pure Mathematics ◽  
And Algebra ◽  
Fundamental Theorems

The superunified field theory consists of a row of discoveries in the realm of pure mathematics. It is two centuries ago that Karl Gauss unified higher arithmetic (number theory), algebra and geometry into what is called pure mathematics. The latter, however, still remains without its fundamental theorem despite that arithmetic and algebra, or even analysis, have their own.

The Complexity of Mathematics

2020 ◽  
Author(s):  
Frank Vega
Keyword(s):  
Number Theory ◽  
Complexity Theory ◽  
Riemann Hypothesis ◽  
Open Problems ◽  
Mathematical Proofs ◽  
Pure Mathematics ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Goldbach’S Conjecture ◽  
Goldbach's Conjecture

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. We prove the Riemann hypothesis using the Complexity Theory. Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. The Goldbach's conjecture is one of the most important and unsolved problems in number theory. Nowadays, it is one of the open problems of Hilbert and Landau. We show the Goldbach's conjecture is true using the Complexity Theory as well. An important complexity class is 1NSPACE(S(n)) for some S(n). These mathematical proofs are based on if some unary language belongs to 1NSPACE(S(log n)), then the binary version of that language belongs to 1NSPACE(S(n)) and vice versa.

Number Theory and Public-Key Cryptography

1991 ◽  
Vol 84 (1) ◽  
pp. 54-63
Author(s):  
Phyllis Lefton
Keyword(s):  
High School ◽  
Number Theory ◽  
High School Students ◽  
Real World ◽  
Mathematics Teacher ◽  
Public Key Cryptography ◽  
Public Key ◽  
School Students ◽  
Pure Mathematics ◽  
Real World Problems

Secret ciphers fascinate students and are an ideal subject with which to motivate them to study mathematics. Recent articles in the Mathematics Teacher, for example, have described how various types of ciphers can be taught to high school students to enhance their learning of statistics (Feltman 1979), matrices (Camp 1985), functions and their inverses (Reagan 1986), and remainder arithmetic (Meneeley 1981). Ciphering also offers interesting examples of how so-called “pure” mathematics can be applied to real-world problems (Hilton 1984).

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