Showing papers on "Prime number published in 1983"
89 citations
1 Jan 1983
TL;DR: This paper shows that similar asymptotic behavior can be obtained for the logarithm problem when q = p m, in the case that m grows with p fixed.
Abstract: The Merkte-Adleman algorithm computes discrete logarithms in GF (q),the finite field with q elements, in subexponential time, when q is a prime number p This paper shows that similar asymptotic behavior can be obtained for the logarithm problem when q = p m , in the case that m grows with p fixed A method of partial precomputation, applicable to either problem, is also presented The precomputation is particularly useful when many logarithms need to be computed for fixed values of p and m
48 citations
TL;DR: In this paper, the Laplace-Beltrami operator and the Selberg zeta-function are applied to the hyperbolic lattice-point problem for cocompact groups.
Abstract: CONTENTS Introduction Chapter I. Spectral theory of the Laplace-Beltrami operator and the Selberg zeta-function § 1. Preliminary information § 2. Discrete groups and the Laplace-Beltrami operator on § 3. The resolvent kernel § 4. The trace of the iterated resolvent for cocompact groups § 5. Huber's theorem § 6. The Selberg zeta-function Chapter II. Applications of the Selberg zeta-function § 7. Prime number theorems § 8. Applications of the prime number theorems § 9. The hyperbolic lattice-point problem for cocompact groups § 10. The hyperbolic lattice-point problem for groups of finite covolume Chapter III. Siegel's theory of quadratic forms and the groups § 11. Conjugacy classes in certain integral orthogonal groups § 12. Some particular examples References
25 citations
TL;DR: In this article, the current status of the Fermat numbers F52, F931, F6835, and F9448 is given, and a new factor of F75 discovered by Gostin is presented.
Abstract: A new factor is given for each of the Fermat numbers F52, F931, F6835, and F9448. In addition, a factor of F75 discovered by Gary Gostin is presented. The current status for all F, is shown in a table. Primes of the form k 2" + 1, k odd, are listed for 31 < k : 149, 1500 < ii ? 4000, and for 151 k 199, 1000 < ii < 4000. Some primes for even larger values of ii are included, the largest one being 5 213165 + 1. Also, a survey of several related questions is given. In particular, values of k such that k 2" + I is composite for every ii are considered, as well as odd values of h such that 3h 2" _1 never yields a twin prime pair.
23 citations
Patent•
20 Jun 1983TL;DR: In this article, a pseudo-random byte is obtained by adding a prime number once or several times to an 8-bit character, depending on whether the preceding addition produced a result larger than or less than 256.
Abstract: Prior systems for generating pseudo-random bits utilize bit-base fedback shift registers. Arrangements operating on a multi-bit, more specifically on a byte-basis, such as microprocessors are not, however, so suitable for this specific use. The invention provide a method with which random bytes, for example, are generated. Such a random byte is obtained by adding a prime number once or several times to an 8-bit character. The choice whether a prime number is added once or several times to the 8-bit character depends on whether the preceding addition produced a result larger than or less than 256. The character obtained (modulo 256) also provides the following 8-bit character. A sequence of bits consisting of several bytes is assembled from random bytes thus generated, each byte being generated on the basis of a different prime number.
15 citations
TL;DR: In this article, the authors proved the following theorems: the letter p always denotes a prime number; P2 represents a number with precisely two prime factors; and the letter n always denotes an integer number; and
Abstract: We write e(x) for e2πix, ∥x∥ for the distance of x from the nearest integer and use A ≫ B to mean |A|
15 citations
TL;DR: A simple technique for performing mod (or modulus) M operations, where M is a prime number, is described and can be applied to the prime memory system as described by Lawrie and Vora.
Abstract: A simple technique for performing mod (or modulus) M operations, where M is a prime number, is described. This technique can be applied to the prime memory system as described by Lawrie and Vora [1].
8 citations
TL;DR: In this article, the ergodic and intermixing theorems of Yu. V. Linnik (Mat. Sb.,43, No. 2, 257-276) are generalized to arbitrary positive quadratic forms.
Abstract: The ergodic and the intermixing theorems of Yu. V. Linnik (Mat. Sb.,43, No. 2, 257–276) are generalized to arbitrary positive quadratic forms
of genus of
, where Ω > 1 is an odd number, of invariants [Ω,1], defined by the character
for all prime numbers ρ¦Ω. One obtains estimates for the remainder term. The method of proof is simplified.
4 citations
TL;DR: It is proved that every vertex-transitive graph can be expressed as the edge-disjoint union of symmetric graphs, including Cayley graphs, multidimensional circulants, and vertex- transitive graphs with a prime or twice a prime number of nodes.
Abstract: In this paper, we prove that every vertex-transitive graph can be expressed as the edge-disjoint union of symmetric graphs. We define a multicycle graph and conjecture that every vertex-transitive graph can be expressed as the edge-disjoint union of multicycles. We verify this conjecture for several subclasses of vertex-transitive graphs, including Cayley graphs, multidimensional circulants, and vertex-transitive graphs with a prime or twice a prime number of nodes. We conclude with some open questions of interest.
4 citations
TL;DR: In this paper, the authors study the E-Divisibi ity br c secutive i tegers and show that if the number of variables is much greater than or equal to the size of the largest variable, then there is a E-divisibity such that (1) 1 = =.-a d (+) 1,1.
Abstract: 1. Thr ugh ut this paper we put e 2 iz =e(a). We write {a}=a-[a] a d ~ a , = =mi ({a}, I-{a}) (i .e ., 1 a ; de tes the dista ce fr m a t the earest i teger). te p sitive abs ute c sta ts. We may say that the p sitive rea umber b is a m st divisib e by the p sitive rea umber a if b is \"sma \". M re exact y, we may say that if r>-O. b O, a positive integer satisfying n>no (E), n°-_t
TL;DR: The relationship between the serial product representing a pattern and the phenomenal experience of a pattern is explored and the factorization is utilized in abstracting a schema, in approximating a feature count model, and in presenting a strategy for holistic vs sequential pattern processing.
Abstract: Prime numbers are used to code various dimensions of an input matrix (receptor surface), i.e., prime numbers code the position of each cell in the matrix, the position of each column and row constituted by the cells of the matrix, and the orientation of each such column or row. The coding permits any pattern or stimulus configuration to be changed into a single, unique number, viz., the serial product of the prime numbers which code the relevant dimensions of the pattern. Storage of a pattern is effected by storage of the serial product. By factoring the serial product, the pattern or stimulus configuration is analyzed into the dimensions (features) specified by the code. The factorization is also utilized in abstracting a schema, in approximating a feature count model, and in presenting a strategy for holistic vs sequential pattern processing. Finally, the relationship between the serial product representing a pattern and the phenomenal experience of a pattern is explored.
1 Jan 1983
TL;DR: In this article, it was shown that if A has infinitely many height one prime ideals, then every ideal in B of grade at least two in the polynomial ring A[t] contains a prime element.
Abstract: Let A be a Krull domain having infinitely many height one primes. It is shown that any ideal of height two in the polynomial ring A[t] contains a prime element. An application to the construction of Dedekind domains with specified class groups is given, along with an example to show the necessity of assuming infinitely many height one primes. Introduction. Krull domains can be viewed as generalizations of factorial rings and, inversely, factorial rings can be characterized as Krull domains with trivial divisor class group. As is well known, an integral domain is factorial if and only if every one of its nonzero prime ideals contains a nonzero principal prime ideal, i.e. a prime element. Thus, to assert the factoriality of an integral domain is to make a statement about the distribution of its prime elements. Factorial rings are those in which the prime elements are in some sense widely distributed. The purpose of this note is to present a result on the distribution of prime elements in integral domains of the form BA[t], where A is a Krull domain and t is an indeterminate. We shall prove that if A has infinitely many height one prime ideals, then every ideal in B of grade at least two contains a prime element. This theorem provides, for polynomial rings over Krull domains, a converse to the obvious fact that in any integral domain an ideal which properly contains a principal prime ideal has grade at least two. We shall give an example to show the necessity of the assumption that A has infinitely many height one prime ideals. Note that this assumption is equivalent to the assertion that A is not a semilocal principal ideal domain.
TL;DR: In this article, it was shown that the Sylow l-subgroup of the ideal class group of a cyclic Galois extension of the rational numbers Q of degree l, where l is a prime number, is elementary abelian when hl = ls and there are s + 1 (or s + 2 if K is real quadratic) ramified primes.
TL;DR: This work enumerates, up to isomorphism, several classes of labeled vertex-transitive digraphs with a prime number of vertices with the aim of determining theorems of vertex-to- vertices correspondence.