Showing papers on "Prime number published in 1972"
Book•
1 Mar 1972
TL;DR: This research proposes a mathematical formula for the distribution of prime numbers, suggesting they follow a regular pattern starting from 2, and also identifies a formula for complex numbers, with potential applications in cryptography and number theory.
Abstract: This research explores the distribution of prime numbers, which are a fundamental topic in number theory. The study originated from the author's fascination with mathematics and the desire to discover something novel. The research proposes that the distribution of prime numbers follows a regular pattern starting from the number 2. The author suggests that prime numbers can be obtained by dividing certain even numbers that have four factors by the number 2, resulting in prime numbers in sequential order. This hypothesis was tested and confirmed through the practical application of the proposed mathematical formula. Additionally, the study found that even numbers greater than or equal to 8, with six or more factors, produce complex numbers. Thus, this research provides two main contributions: firstly, a mathematical formula for the distribution of prime numbers, and secondly, a formula for the distribution of complex numbers. These findings have potential applications in various mathematical fields, including cryptography and problem-solving in number theory.
211 citations
TL;DR: The primary object of this paper is the development of machinery which simplifies this remaining calculation of the weight distributions for this whole sequence of codes.
Abstract: With any fixed prime number p and positive integer N , not divisible by p , there is associated an infinite sequence of cyclic codes. In a previous article it was shown that a theorem of Davenport-Hasse reduces the calculation of the weight distributions for this whole sequence of codes to a single calculation (essentially that of calculating the weight distribution for the simplest code of the sequence). The primary object of this paper is the development of machinery which simplifies this remaining calculation. Detailed examples are given. In addition, tables are presented which essentially solve the weight distribution problem for all such binary codes with N
178 citations
TL;DR: In this paper, the authors give a simple proof of the equivalence of two statements: (1) Γ is a symmetric graph with p vertices, each having degree n ≥ 1; (2) the integer n is an even divisor of p − 1; and (3) the graph is isomorphic to the graph whose vertices are the elements of Z(p) and whose edges are the pairs { a, a+h } where a ∈ Z( p) and h ∈ H, the unique subgroup of Z *(
Abstract: A graph Γ is called symmetric if its automorphism group is transitive on its vertices and edges. Let p be an odd prime, Z(p) the field of integers modulo p , and Z *( p ) = ( a ∈ Z ( p ) | a ≠ 0}, the multiplicative subgroup of Z(p) . This paper gives a simple proof of the equivalence of two statements: (1) Γ is a symmetric graph with p vertices, each having degree n ≥ 1; (2) the integer n is an even divisor of p − 1 and Γ is isomorphic to the graph whose vertices are the elements of Z(p) and whose edges are the pairs { a , a+h } where a ∈ Z(p) and h ∈ H , the unique subgroup of Z *( p ) of order n . In addition, the automorphism group of Γ is determined.
27 citations
Book•
1 Jan 1972
26 citations
11 citations
TL;DR: In this article, the authors pointed out the close relationship of cyclic graphs and cyclic designs and developed a general constructive method for enumerating cyclic paired-comparison designs.
9 citations
5 citations
TL;DR: In this article, the authors discuss algebraic function fields and nonstandard arithmetic, and present a nonstandard model of the field of rational numbers Q for a higher order language, where * Q may be an ultra power of Q and an element of * Q which is not contained in Q.
Abstract: Publisher Summary This chapter discusses the algebraic function fields and nonstandard arithmetic. Let * Q be a nonstandard model of the field of rational numbers Q , for a higher order language. In particular, * Q may be an ultra power of Q . Let (a) be an element of * Q which is not contained in Q . Then it is easy to verify that (a) is transcendental over Q . Thus, the field A = Q (a) c * Q is the field of rational functions with rational coefficients. An internal valuation of * Q is given either by a nonstandard prime number in * Q or by a standard prime number in Q (and * Q ) or by the Archimedean valuation of * Q .
4 citations
1 Jan 1972
TL;DR: In this article, all primes of the form 2A3n + 1 and 2A 3n - 1, where 1? A? 50 and 1 < n < 325, are found.
Abstract: All primes of the form 2A3n + 1 and of the form 2A3n - 1, where 1 ? A ? 50 and 1 < n < 325, are found. Some large twin primes are also determined.
2 citations
1 citations
Journal Article•
TL;DR: In this article, a simplication of the original Malliavin argument using Leibniz's rule has been proposed, which replaces an interpolation argument of Mallian by an argument using a better trigonometric inequality.
TL;DR: In this paper, a history of the problem is given, including superperiod data for quadratic and cubic nonlinear terms, together with a computation for a prime number of particles in the string extension to a circular array is discussed.