Showing papers on "Prime number published in 1982"
Book•
1 Jul 1982
TL;DR: This chapter discusses the Factorization of Integers, a branch of Arithmetic Functions, which addresses the problem of how to estimate the number of factors in a discrete-time system.
Abstract: 1. The Factorization of Integers.- 1.1 Divisibility.- 1.2 Prime Numbers and Composite Numbers.- 1.3 Prime Numbers.- 1.4 Integral Modulus.- 1.5 The Fundamental Theorem of Arithmetic.- 1.6 The Greatest Common Factor and the Least Common Multiple.- 1.7 The Inclusion-Exclusion Principle.- 1.8 Linear Indeterminate Equations.- 1.9 Perfect Numbers.- 1.10 Mersenne Numbers and Fermat Numbers.- 1.11 The Prime Power in a Factorial.- 1.12 Integral Valued Polynomials.- 1.13 The Factorization of Polynomials.- Notes.- 2. Congruences.- 2.1 Definition.- 2.2 Fundamental Properties of Congruences.- 2.3 Reduced Residue System.- 2.4 The Divisibility of 2p-1-1 by p2.- 2.5 The Function ?(m).- 2.6 Congruences.- 2.7 The Chinese Remainder Theorem.- 2.8 Higher Degree Congruences.- 2.9 Higher Degree Congruences to a Prime Power Modulus.- 2.10 Wolstenholme's Theorem.- 3. Quadratic Residues.- 3.1 Definitions and Euler's Criterion.- 3.2 The Evaluation of Legendre's Symbol.- 3.3 The Law of Quadratic Reciprocity.- 3.4 Practical Methods for the Solutions.- 3.5 The Number of Roots of a Quadratic Congruence.- 3.6 Jacobi's Symbol.- 3.7 Two Terms Congruences.- 3.8 Primitive Roots and Indices.- 3.9 The Structure of a Reduced Residue System.- 4. Properties of Polynomials.- 4.1 The Division of Polynomials.- 4.2 The Unique Factorization Theorem.- 4.3 Congruences.- 4.4 Integer Coefficients Polynomials.- 4.5 Polynomial Congruences with a Prime Modulus.- 4.6 On Several Theorems Concerning Factorizations.- 4.7 Double Moduli Congruences.- 4.8 Generalization of Fermat's Theorem.- 4.9 Irreducible Polynomials mod p.- 4.10 Primitive Roots.- 4.11 Summary.- 5. The Distribution of Prime Numbers.- 5.1 Order of Infinity.- 5.2 The Logarithm Function.- 5.3 Introduction.- 5.4 The Number of Primes is Infinite.- 5.5 Almost All Integers are Composite.- 5.6 Chebyshev's Theorem.- 5.7 Bertrand's Postulate.- 5.8 Estimation of a Sum by an Integral.- 5.9 Consequences of Chebyshev's Theorem.- 5.10 The Number of Prime Factors of n.- 5.11 A Prime Representing Function.- 5.12 On Primes in an Arithmetic Progression.- Notes.- 6. Arithmetic Functions.- 6.1 Examples of Arithmetic Functions.- 6.2 Properties of Multiplicative Functions.- 6.3 The Mobius Inversion Formula.- 6.4 The Mobius Transformation.- 6.5 The Divisor Function.- 6.6 Two Theorems Related to Asymptotic Densities.- 6.7 The Representation of Integers as a Sum of Two Squares.- 6.8 The Methods of Partial Summation and Integration.- 6.9 The Circle Problem.- 6.10 Farey Sequence and Its Applications.- 6.11 Vinogradov's Method of Estimating Sums of Fractional Parts.- 6.12 Application of Vinogradov's Theorem to Lattice Point Problems.- 6.13 ?-results.- 6.14 Dirichlet Series.- 6.15 Lambert Series.- Notes.- 7. Trigonometric Sums and Characters.- 7.1 Representation of Residue Classes.- 7.2 Character Functions.- 7.3 Types of Characters.- 7.4 Character Sums.- 7.5 Gauss Sums.- 7.6 Character Sums and Trigonometric Sums.- 7.7 From Complete Sums to Incomplete Sums.- 7.8 Applications of the Character Sum $$\sum\limits_{x = 1}^p {\left( {\frac{{x^2 + ax + b}}{p}} \right)} $$.- 7.9 The Problem of the Distribution of Primitive Roots.- 7.10 Trigonometric Sums Involving Polynomials.- Notes.- 8. On Several Arithmetic Problems Associated with the Elliptic Modular Function.- 8.1 Introduction.- 8.2 The Partition of Integers.- 8.3 Jacobi's Identity.- 8.4 Methods of Representing Partitions.- 8.5 Graphical Method for Partitions.- 8.6 Estimates for p(n).- 8.7 The Problem of Sums of Squares.- 8.8 Density.- 8.9 A Summary of the Problem of Sums of Squares.- 9. The Prime Number Theorem.- 9.1 Introduction.- 9.2 The Riemann ?-Function.- 9.3 Several Lemmas.- 9.4 A Tauberian Theorem.- 9.5 The Prime Number Theorem.- 9.6 Selberg's Asymptotic Formula.- 9.7 Elementary Proof of the Prime Number Theorem.- 9.8 Dirichlet's Theorem.- Notes.- 10. Continued Fractions and Approximation Methods.- 10.1 Simple Continued Fractions.- 10.2 The Uniqueness of a Continued Fraction Expansion.- 10.3 The Best Approximation.- 10.4 Hurwitz's Theorem.- 10.5 The Equivalence of Real Numbers.- 10.6 Periodic Continued Fractions.- 10.7 Legendre's Criterion.- 10.8 Quadradic Indeterminate Equations.- 10.9 Pell's Equation.- 10.10 Chebyshev's Theorem and Khintchin's Theorem.- 10.11 Uniform Distributions and the Uniform Distribution of n? (mod 1).- 10.12 Criteria for Uniform Distributions.- 11. Indeterminate Equations.- 11.1 Introduction.- 11.2 Linear Indeterminate Equations.- 11.3 Quadratic Indeterminate Equations.- 11.4 The Solution to ax2 + bxy + cy2=k.- 11.5 Method of Solution.- 11.6 Generalization of Soon Go's Theorem.- 11.7 Fermat's Conjecture.- 11.8 Markoff's Equation.- 11.9 The Equation x3 + y3 + z3 + ?3=0.- 11.10 Rational Points on a Cubic Surface.- Notes.- 12. Binary Quadratic Forms.- 12.1 The Partitioning of Binary Quadratic Forms into Classes.- 12.2 The Finiteness of the Number of Classes.- 12.3 Kronecker's Symbol.- 12.4 The Number of Representations of an Integer by a Form.- 12.5 The Equivalence of Formsmod q.- 12.6 The Character System for a Quadratic Form and the Genus.- 12.7 The Convergence of the Series K(d).- 12.8 The Number of Lattice Points Inside a Hyperbola and an Ellipse.- 12.9 The Limiting Average.- 12.10 The Class Number: An Analytic Expression.- 12.11 The Fundamental Discriminants.- 12.12 The Class Number Formula.- 12.13 The Least Solution to Pell's Equation.- 12.14 Several Lemmas.- 12.15 Siegel's Theorem.- Notes.- 13. Unimodular Transformations.- 13.1 The Complex Plane.- 13.2 Properties of the Bilinear Transformation.- 13.3 Geometric Properties of the Bilinear Transformation.- 13.4 Real Transformations.- 13.5 Unimodular Transformations.- 13.6 The Fundamental Region.- 13.7 The Net of the Fundamental Region.- 13.8 The Structure of the Modular Group.- 13.9 Positive Definite Quadratic Forms.- 13.10 Indefinite Quadratic Forms.- 13.11 The Least Value of an Indefinite Quadratic Form.- 14. Integer Matrices and Their Applications.- 14.1 Introduction.- 14.2 The Product of Matrices.- 14.3 The Number of Generators for Modular Matrices.- 14.4 Left Association.- 14.5 Invariant Factors and Elementary Divisors.- 14.6 Applications.- 14.7 Matrix Factorizations and Standard Prime Matrices.- 14.8 The Greatest Common Factor and the Least Common Multiple.- 14.9 Linear Modules.- 15. p-adic Numbers.- 15.1 Introduction.- 15.2 The Definition of a Valuation.- 15.3 The Partitioning of Valuations into Classes.- 15.4 Archimedian Valuations.- 15.5 Non-Archimedian Valuations.- 15.6 The ?-Extension of the Rationals.- 15.7 The Completeness of the Extension.- 15.8 The Representation of p-adic Numbers.- 15.9 Application.- 16. Introduction to Algebraic Number Theory.- 16.1 Algebraic Numbers.- 16.2 Algebraic Number Fields.- 16.3 Basis.- 16.4 Integral Basis.- 16.5 Divisibility.- 16.6 Ideals.- 16.7 Unique Factorization Theorem for Ideals.- 16.8 Basis for Ideals.- 16.9 Congruent Relations.- 16.10 Prime Ideals.- 16.11 Units.- 16.12 Ideal Classes.- 16.13 Quadratic Fields and Quadratic Forms.- 16.14 Genus.- 16.15 Euclidean Fields and Simple Fields.- 16.16 Lucas's Criterion for the Determination of Mersenne Primes.- 16.17 Indeterminate Equations.- 16.18 Tables.- Notes.- 17. Algebraic Numbers and Transcendental Numbers.- 17.1 The Existence of Transcendental Numbers.- 17.2 Liouville's Theorem and Examples of Transcendental Numbers.- 17.3 Roth's Theorem on Rational Approximations to Algebraic Numbers.- 17.4 Application of Roth's Theorem.- 17.5 Application of Thue's Theorem.- 17.6 The Transcendence of e.- 17.7 The Transcendence of ?.- 17.8 Hilbert's Seventh Problem.- 17.9 Gelfond's Proof.- Notes.- 18. Waring's Problem and the Problem of Prouhet and Tarry.- 18.1 Introduction.- 18.2 Lower Bounds for g(k) and G(k).- 18.3 Cauchy's Theorem.- 18.4 Elementary Methods.- 18.5 The Easier Problem of Positive and Negative Signs.- 18.6 Equal Power Sums Problem.- 18.7 The Problem of Prouhet and Tarry.- 18.8 Continuation.- 19. Schnirelmann Density.- 19.1 The Definition of Density and its History.- 19.2 The Sum of Sets and its Density.- 19.3 The Goldbach-Schnirelmann Theorem.- 19.4 Selberg's Inequality.- 19.5 The Proof of the Goldbach-Schnirelmann Theorem.- 19.6 The Waring-Hiibert Theorem.- 19.7 The Proof of the Waring-Hiibert Theorem.- Notes.- 20. The Geometry of Numbers.- 20.1 The Two Dimensional Situation.- 20.2 The Fundamental Theorem of Minkowski.- 20.3 Linear Forms.- 20.4 Positive Definite Quadratic Forms.- 20.5 Products of Linear Forms.- 20.6 Method of Simultaneous Approximations.- 20.7 Minkowski's Inequality.- 20.8 The Average Value of the Product of Linear Forms.- 20.9 Tchebotaref's Theorem.- 20.10 Applications to Algebraic Number Theory.- 20.11 The Least Value for |?|.
1,067 citations
74 citations
64 citations
TL;DR: It is shown that 17 is also not possible by reducing the problem to the consideration of 16 and finding a weight 12 vector, by computer, in each of these codes.
Abstract: It is an interesting open question whether an extremal (72, 36, 16) doubly even code C exists. In [3] the odd prime numbers which can divide the order of the group of C were determined. The largest of these, 23, was eliminated by finding weight 12 vectors in 384 codes [8]. The next largest prime remaining is 17. It is shown that 17 is also not possible by reducing the problem to the consideration of 16.17^{3} codes and then finding a weight 12 vector, by computer, in each of these codes.
30 citations
26 citations
TL;DR: An interesting open question is whether a (72, 36, 16) doubly even code C exists, which is shown to have minimum weight 12 or less by reducing the problem to the consideration of 348 codes.
Abstract: An interesting open question is whether a (72, 36, 16) doubly even code C exists In [3] the odd prime numbers which can divide the order of the group of C were determined and 23 is the largest of these Twenty-three is eliminated by reducing the problem to the consideration of 348 codes, each of which is shown to have minimum weight 12 or less One of these codes, denoted by C' , arises from the (a + x, b + x, a + b + x) construction where a and b are in one quadratic residue code and x is in the other The weight distribution of C' is given
24 citations
TL;DR: The eigensystem for the Fast Fourier transform, FFT, known for several years, can be used to design FFT algorithms and it is found that for every prime number transform there are only 4 distinct eigenvalues.
23 citations
TL;DR: In this article, an asymptotic formula for the number of representations of a natural number as the sum of the squares of four integers, two of which are prime, is given.
Abstract: In the article an asymptotic formula is determined for the number of representations of a natural number as the sum of the squares of four integers, two of which are prime. Bibliography: 25 titles.
23 citations
1 Jan 1982
16 citations
TL;DR: Chebyshev as discussed by the authors showed that (∗) is only a special case (in several respects) of the Riemann hypothesis and showed that it can be generalized to a wider class of progressions.
8 citations
TL;DR: In this article, the distribution of integers and prime numbers in sequences of the formF¯¯¯¯c1∩F¯¯c2 withc>1 was investigated. But the distribution was not restricted to integers.
Abstract: The distribution of integers and prime numbers in sequences of the formF
c1∩F
c2 is investigated. HereF
c={[n
c]:n∈ ℕ} withc>1.
1 Jan 1982
TL;DR: In this article, it was shown that the Stickelberger ideal of k can be expressed in terms of the relative class number of a cyclotomic field, which is the ratio of the first factor of the class number to the number of distinct prime numbers in the complex conjugation of k.
Abstract: Let k be any imaginary abelian fieldX R the integral group ring of G = Gal(k/(2) and S the Stickelberger ideal of k. Roughly speakingX the relative class number hof k is expressed as the index of S in a certain ideal A of R described by means of G and the complex conjugation of k; c-h= [A: S] with a rational number cin -NJ = {n/2;n e NJ} which can be described without hand is of lower than hif the conductor of k is sufficiently large (cf. [6 9 1O]; see also [5]). We shall prove that 2ca natural numberX divides 2([k: (2]/2)lk 1/2. In particularX if k varies through a sequence of imaginary abelian fields of degrees boundedX then ctakes only a finite number of values. On the other handX it will be shown that ccan take any value in 2NJ when k ranges over all imaginary abelian fields. In this connectionX we shall also make a simple remark on the divisibility for the relative class number of cyclotomic fields. Let 2, C;!!, 1R, and C denote the rational integer ring, the rational number field, the real number field, and the complex number field, respectively. A finite abelian extension over C;!! contained in C will be called an abelian field. Let k be an imagi- nary abelian field, namely, an abelian field not contained in 1R. We denote by R(k) the group ring of the Galois group G = Gal(k/C;!!) over z and by s(H), for any subgroup H of G, the sum in R(k) of all elements in H. Put A(k) = { E R(k); (1 + jk)°l = as(G) for some a E E}, where ik denotes the complex conjugation of k. Let hk denote the relative class number of k (i.e., the so-called first factor of the class number of k), Qk the unit index of k, 9k the number of distinct prime numbers ramified in k, and S(k) the Stickelberger ideal of k in the sense of Iwasawa-Sinnott, which is an additive sub- group of A(k) with finite index (for the definition of the Stickelberger ideal, see [6, 10]). We define ck as the ratio of the index [A(k): S(k)] to hk: Ck hk = [A(k): S(k)] The product QkCk is known to be a natural number and is determined by Sinnott in various cases, for example, in the case 9k = 1 or 2 (cf. [10]). He has also shown in [9] that, if k is a cyclotomic field, then Ck = 2b where b = 0 or 29k-1-1 according as 9k = 1 or 9k > 2 (for the case 9k = 1, see [6]). In this paper, we shall give an additional result concerning the range of Ck . Received by the editors July 31 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary llR20 llR29; Secondary llN25 llR18.
TL;DR: Some recent results in number theory, discovered with electronic computers, are discussed in this article, and many open questions, unsolved problems, and conjectures in these areas are mentioned, illustrating the tremendous impact of computers on pure mathematical research.
Abstract: Some recent results in number theory, discovered with electronic computers, are discussed in this paper. Central results, and results closely related to them, have been selected from the following areas: prime, composite, perfect numbers; distribution of primes, twin primes; Fermat's and Wilson's quotients; Fibonacci and pseudo‐Fibonacci numbers; and zeros of the Riemann zeta function. Many open questions, unsolved problems, and conjectures in these areas are mentioned. By discussing some of the recent computer‐established results in number theory, this paper illustrates the tremendous impact of computers on pure mathematical research.
1 Mar 1982
TL;DR: In this paper, the authors characterized polynomials over GF(q) which commute with a translation by an element of the field, i.e., the polynomial f(x) with coefficients in GF(k) for which f(k + a) = k(x + k + a).
Abstract: Let p be a prime number, q = pn for some positive integer n, and GF(q) the finite field with q elements Let a be a nonzero element of GF(q) In [3] Wells characterized those polynomials over GF(q) which commute with a translation by an element of the field, ie, he characterized those polynomials f(x) with coefficients in GF(q) for which f(x + a) = f(x) + a In the present paper we characterize those polynomials f(x) for which
TL;DR: In this paper, it was shown that Fermat's theorem on primes may be proved using symmetry properties of Ising spin configurations; and that similarly this may be extended to certain composite numbers.
Abstract: The prime numbers play a central role in the theory of numbers. We show that Fermat’s theorem on primes may be proved using symmetry properties of Ising‐spin configurations; and that similarly this may be extended to certain composite numbers. Our method of proof suggests a ’’physical’’ interpretation of the primes.
1 Feb 1982
TL;DR: In this paper, the spectrum of a complete algebraic variety has one of the following values S o, S 1, S m, S p, where m∈ω; p is a prime number.
Abstract: Publisher Summary This chapter discusses the number of models in complete varieties. A class M of algebraic systems of a signatureΣ, closed with respect to subsystems, Cartesian products, and homomorphic images is called a “variety.” A variety M is called complete if there exist infinite M-systems and all of them are elementary equivalent. All the varieties considered have signature of finite or countable power. If K is a class of algebraic systems, then by S K (x) the number is denoted of isomorphic types of K-systems of power x . Such a correspondence S is called as the “spectrum” of K. The chapter describes all the spectra S for complete varieties M. This description is obtained from the theorem that explains that spectrum of a complete variety has one of the following values S o , S 1 , S m , S p , where m∈ω; p is a prime number.
1 Jan 1982
TL;DR: In this paper, it was shown that cyclotomic fields have always occupied an eminent place in Iwasawa theory and it is easiest to begin with this particular and interesting case.
Abstract: Cyclotomic fields have always occupied an eminent place in Iwasawa theory and it is easiest to begin with this particular and interesting case. Put ζm = exp (πi/m) and let p be a prime number.
TL;DR: Given any integer t ≥ 2 and any prime number p, a graph Γp,t is constructed whose adjacency matrix is nilpotent of index t over Zp' the field of p elements.
Abstract: Given any integer t ≥ 2 and any prime number p, a graph Γp,t is constructed whose adjacency matrix is nilpotent of index t over Zp' the field of p elements.
TL;DR: Several different ways in which computers have aided in the growth of various branches of number theory are described.
Abstract: 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.
TL;DR: In this paper, a simple extension of Vaughan's method is presented, which is essentially as powerful as any of the techniques mentioned above, to discuss its general implications, and to apply it to the proof of the following result of Huxley [4], which has previously only been within the scope of the zero density method.
Abstract: 1. Introduction. Many problems involving prime numbers depend on estimating sums of the form ΣΛ(n)f(n), for appropriate functions f(n), (here, as usual, Λ(n) is the von Mangoldt function). Three distinct general methods have been used to estimate such sums. The earliest is due to Vinogradov (see [13, Chapter 9]); the second involves zerodensity bounds for Dirichlet L–functions (see [8, Chapters 15 and 16] for example); and the third, due to Vaughan (see [12] for example) uses an arithmetical identity as will be explained later. The second and third methods are much simpler to apply than the first. On the other hand Vinogradov's technique is at least as powerful as Vaughan's and occasionally more so. In many cases Vaughan's identity yields better bounds than the use of zero–density estimates, but sometimes they are worse. The object of this paper is to present a simple extension of Vaughan's method which is essentially as powerful as any of the techniques mentioned above, to discuss its general implications, and to apply it to the proof of the following result of Huxley [4], which has previously only been within the scope of the zero density method.
TL;DR: In this paper, the authors present a table of table of tables of this paper : Table of Table 1.3.1.1-3.2.0.1]
Abstract: Table of