Showing papers on "Prime number published in 1974"
TL;DR: In this paper, it was shown that a homomorphism Z of G into the multiplicative group of Qv can be seen as a ray class character in class field theory, and the work of Klingen and Siegel [9] enables us to associate with Z an algebraic number Ls 0(, n ) in Qp as follows.
Abstract: Introduction Let p be a prime number, and let Zp, Qp, and ~p denote the ring of p-adic integers, the field of p-adic numbers, and an algebraic closure ofQp, respectively. Let K be a totally real number field of finite degree over the rational field Q, and M a finite abelian extension of K. Write S for the set of primes of K lying above p, and G for the Galois group of M over K. Suppose we are given a homomorphism Z of G into the multiplicative group of Qv. For each integer n>0, the work of Klingen and Siegel [9] enables us to associate with Z an algebraic number Ls0(, n ) in Qp as follows. By class field theory, we can view X as a ray class character. Let be the conductor of X, 9 = f(P), and R~ the ray class group modulo .q. For each r~Ro, let ~(r , s) be the partial zeta function of r (for the precise definition, see w 1). Klingen and Siegel [9] have proven that ~q(r, n ) is always rational, and we define
45 citations
41 citations
TL;DR: A matrix A in the semigroup N n of non-negative n×n matrices is prime if A is not monomial and A =BC,B CeN n implies that either B or C is monomial as discussed by the authors.
Abstract: A matrix A in the semigroup N n of non-negative n×nmatrices is prime if A is not monomial and A=BC,B CeN n implies that either B or C is monomial. One necessary and another sufficient condition are given for a matrix in N n to be prime. It is proved that every prime in N n is completely decomposable.
35 citations
TL;DR: For r? 1 and large N, a well-known conjecture of Hardy and Littlewood implies that the number of primes p n such that p + 2 r is the least prime greater than p is as- ymptotic as discussed by the authors.
Abstract: For r ? 1 and large N, a well-known conjecture of Hardy and Littlewood implies that the number of primes p ? N such that p + 2r is the least prime greater than p is as- ymptotic to
26 citations
12 citations
TL;DR: In this article, the irreducible representations of the generalized Clifford groups Gmn in the case where m is a prime number, are extended to the case when m is any integer.
10 citations
TL;DR: In this article, a necessary and sufficient condition for a function in H to be a prime is established, based on Nevanlinna's theory of meromorphic functions and some elementary facts about algebraic functions.
Abstract: Let H denote the set of all the entire functions f(z) of the form: f(z) = h(z)eM + k(z) where p(z) is a nonconstant polynomial of degree m, and A(# 0), k (# constant) are two entire functions of order less than m. In this paper, a necessary and sufficient condition for a function in H to be a prime is established. Several generalizations of known results follow. Some sufficient conditions for primeness of various subclasses of H are derived. The methods used in the proofs are based on Nevanlinna's theory of meromorphic functions and some elementary facts about algebraic functions.
8 citations
7 citations
7 citations
TL;DR: In this article, it was shown that for positive rational integers, the polynomial x2 + x + A is prime for x = 0, 1, 2, 3, 4, A − 2.
TL;DR: In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal's triangle are reduced modulo m as mentioned in this paper.
Abstract: In a very interesting recent article [1] W. A. Broomhead described an investigation carried out by staff and pupils at Tonbridge School of the patterns which result when the numbers in Pascal’s triangle are reduced modulo m. For the case when m equals a prime number, p, the pattern formed by the zeros in the reduced triangle (corresponding to binomial coefficients divisible p) was completely described and the following result (stated by G. Gilbart-Smith) was proved: In the (n + l)th row of Pascal’s triangle, there are
1 Mar 1974
TL;DR: In this paper, the polynomials over a finite field which commute with translation by an element of the field are characterized and a generalization of a long-known theorem about centralizers of permutations is used in obtaining the characterization.
Abstract: The polynomials over a finite field which commute with translation by an element of the field are characterized. A generalization of a long-known theorem about centralizers of permutations is used in obtaining the characterization. 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). Theorem 1 below characterizes the polynomials f(x) with coefficients in GF(q) for which degf < q 1 and (1) f(x+ a)= f(x)+ a. This will actually characterize all polynomials over GF(q) satisfying (1), since each such polynomial is congruent (modx q x) to a unique such polynomial of degree < q 1. The characterization will be obtained by equating coefficients in (1), but the computation will be shortened by using a generalization of a longknown theorem about centralizers of permutations (Theorem 2). Theorem 1. Let f(x) = bo + b1x + b2x2 + ... + b xu, with br E GF(q) for r = 1, 2, ..., u, and u = q 1. Then f(x) satisfies (1) if and only if
Journal Article•
TL;DR: In this article, it was shown that at least in so far the prime number theorem is an asympototic result, or with mild error term, is concerned, the underlying multiplicative structure is not at all important.
Abstract: Following the earliest proofs of the prime number theorem, and continuing thereafter, various attempts were made to examine the extent to which the methods used were restricted to the natural primes, and to free them from this restriction. De la Vallee Poussin's proofs of the Prime Number Theorem for arithmetic progressions, and bis investigation of quadratic forms, may be seen among the earliest instances of this spirit. There followed the development of various zeta-functions\" continuing to the present time, and the investigation of their analytic behavior and their relationship to the underlying structure generating them — an early success in this line being Landau's Prime Ideal Theorem. Beurling showed how simply viewing the natural primes äs the generators of a commutative semigroup with unit could lead under mild auxiliary conditions to a prime number theorem\", and various investigations of these \"Beurling's generalized primes\" have clarified the relationship between the auxiliary conditions imposed on the semigroup (some kind of \"density\" of its elements) and the error term in the analogue of the prime number theorem obtained (e. g. [1], [2], [3], [11], [15], a complete bibliography up to 1969 may be found in [1]). In all of the results mentioned above, and many others, some kind of multiplicativity analogous to the generation of the natural numbers by the natural primes lay at the base of the discussion. One of the main purposes of this paper is to demonstrate that at least in so far äs the prime number theorem äs an asympototic result, or with mild error term, is concerned, the underlying multiplicative structure is not at all important. We obtain prime number theorem analogues for the following Situation: Suppose h (n) is a bounded arithmetic function which has some other nice\" properties (see Theorem 3 below for an explicit Statement), and let h(n) be the coefficient of the ordinary Dirichlet series
1 Jan 1974
TL;DR: In this paper, Delange-Pisot-Poitou et al. implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions).
Abstract: © Séminaire Delange-Pisot-Poitou. Théorie des nombres (Secrétariat mathématique, Paris), 1973-1974, tous droits réservés. L’accès aux archives de la collection « Séminaire Delange-Pisot-Poitou. Théorie des nombres » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
1 Jan 1974
TL;DR: In this paper, it was shown that Dirichlet's theorem on primes in an arithmetic progression is equivalent to the statement that every unit of a certain quotient ring Z of the nonstandard integers is the image of an infinite prime.
Abstract: It is shown that Dirichlet's theorem on primes in an arithmetic progression is equivalent to the statement that every unit of a certain quotient ring Z of the nonstandard integers is the image of an infinite prime. The ring Z is the completion of Z relative to the "natural" topology on Z. 1. Notation. Throughout this note A shall denote the natural numbers, Z the rational integers, and P the positive primes. We shall follow the approach of Machover and Hirschfeld, (2), in our use of nonstandard analysis. Thus U is to be a universal set containing N and *U will be a comprehensive (6, p. 446) enlargement of U. The nonstandard natural numbers *A can be expressed as *A=AuAœ where Nw is the set of infinite natural numbers. Similarly, *P=P\JPœ, Px the set of infinite
Journal Article•
TL;DR: In this paper, the conditions générales d'utilisation (http://www.compositio.org/conditions) of the agreement with the Foundation Compositio Mathematica are described.
Abstract: © Foundation Compositio Mathematica, 1974, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
TL;DR: A very compact list of the primes to 100000 is given and applications to computer storage of primes is indicated in this article, where a table gives a compact listing of prime numbers for the interval from zero to 100,000, where each decade (an interval of 10 numbers) can contain at most 4 primes: those numbers congruent to 1, 3, 7 or 9 mod 10.
Abstract: A very compact listing of the primes to 100000 is given and applications to computer storage of primes is indicated. The accompanying table gives a compact listing of prime numbers for the interval from zero to 100000. Each decade (an interval of 10 numbers) can contain at most 4 primes: Those numbers congruent to 1, 3, 7 or 9 mod 10 (with the obvious exception of the first four primes: 2, 3, 5 and 7). There exist j4=o (4) =16 possible combinations for each decade. These 16 possibilities are represented in the table by the symbols 0, A, B, C, ,N and P and their meanings are indicated at the bottom of the table. (An additional symbol, Q, is used for the first four primes.) The left-hand column in the table indicates the thousands, the top row the hundreds and the second row the decades. Thus, for example, to see if 39199 is prime we look at the row for 39000 in the 100 column and down the 9 column and find the symbol G. This shows that in this decade 1 and 9 are primes; i.e., 39191 and 39199 are both primes. In addition, the table lists the counts of primes for each interval of 100 and for each 1000. The key for the hundreds is indicated at the bottom of the first page of the table. Thus, there are P = 25 primes from 0 to 100 and L =21 primes from 100 to 200 etc. From 0 to 1000 there are 168 primes. Also, as shown on the bottom of the table, there are 5133 primes from zero to 50000 and 9592 primes from zero to 100000. Since, in most computers, words are represented in the octal or hexadecimal (16) systems the listing provides an efficient means for computer storage of primes. Queens College, CUNY Flushing, New York 11367 Copyright C 1974, American Mathematical Society 855 Received March 14, 1973. AMS (MOS) subject classifications (1970). Primary 1OA25.
TL;DR: In this paper, it was shown that the prime number theorem can be made plausible by quite simple and elementary methods which, in addition, give a fascinating insight into the stochastic nature of the theorem.
Abstract: The Prime Number Theorem is a remarkable and rather deep result in the theory of numbers. The aim of this article is to show that this theorem can be made plausible by quite simple and elementary methods which, in addition, give a fascinating insight into the stochastic nature of that theorem. The material of this article is intended for use at the college level to integrate and motivate chapters on probability, sequence and series (especially the harmonic series), and the logarithmic function usually treated in most elementary mathematics courses. The heuristic arguments in this article can be tightened up, but that cannot be done at college level. For every real number x let 7r(x) be the number of primes less than or equal to x. One finds that 7r(10) = 4, 7r(100) = 25, 7r(1000) = 168, etc. The function x -* 7r(x) will be called the prime number function. All attempts to find a formula for 7r(x) representing 7r(x) in "closed form" by a finite number of "known" functions have failed, and will necessarily fail. There are, however, some simple asymptotic expressions for 7r(x), such as