Showing papers on "Prime number published in 1987"
TL;DR: A survey of direct and recursive constructions for cyclic Steiner 2-designs is given in this article, which is based on perfect systems of difference sets and additive sequences of permutations.
Abstract: This paper surveys direct and recursive constructions for cyclic Steiner 2-designs. A new method is presented for cyclic designs with blocks having a prime number of elements. Several new constructions are given for designs with block size 4 which are based on perfect systems of difference sets and additive sequences of permutations.
63 citations
TL;DR: An implementation of the Cohen-Lenstra version of the Adleman-Pomerance-Rumely primality test, which can now routinely be proved within approximately ten minutes, is presented.
Abstract: An implementation of the Cohen-Lenstra version (see ibid., vol. 42, p.297-330, 1984) of the Adleman-Pomerance-Rumely primality test (see L.M. Adleman, C. Pomerance and R.S. Rumely, Ann. of Math., vol.117, p.173-206, 1983) is presented. Primality of prime numbers of up to 213 decimal digits can now routinely be proved within approximately ten minutes
61 citations
Book•
1 Jan 1987
TL;DR: The algebra of congruence classes congruences of higher degree is a generalization of the algebra of the reals diophantine equations as discussed by the authors, which is also related to our work.
Abstract: What is number theory? divisibility prime numbers numerical functions the algebra of congruence classes congruences of higher degree the number theory of the reals diophantine equations.
31 citations
TL;DR: For any prime number k ≥ 3 and any commutative ring A, this paper showed that every element of A k is the sum of k 3 k th powers (for any A ) for some k such as k = 11 or 19.
17 citations
1 Apr 1987
TL;DR: In this paper, it was shown that every irreducible rational represention of a finite p-group is the difference of two transitive permutation representations of the group.
Abstract: The following theorem is proved for any prime p: Every irreducible rational represention of a finite p-group is the difference of two transitive permutation representations Also given are two useful results about representations of p-groups, which are known to experts in the field 1 Statement of results Presented here are three results about characters of finite p-groups The first two are probably known to people familiar with the subject The first follows directly from a theorem of Solomon [7, p 156, Theorem 4] and the second from a generalization of this in Feit [2, p 73, 143] Since they do not seem to have been stated and proved explicitly our purpose in doing so here is to make them available to a wider audience The third result generalizes a theorem of Graeme Segal [6] that may not be well known among group theorists As Segal states in his paper, Feit has observed that this theorem may be proved using the result above We present such a proof here and obtain the following result THEOREM An irreducible rational representation of a finite p-group, p a prime, is the difference of two transitive permutation representations Let Q denote the rational field and QG the rational group algebra of a finite group G Let p be a prime number Gal denotes a Galois group [X2 : Xi] denotes the index of X1 in X2 for either fields or groups Absolute value bars denote order All groups are finite here THEOREM 1 Let G be a p-group and X an irreducible complex character of G Then one of the following holds: (i) There exists a linear character A on a subgroup H of G which induces X and generates the same field as X; that is AG = X and Q(A) = Q(X) (ii) p = 2 and there exist subgroups H < K in G with [K : H] = 2 and a linear character A of H such that with AK, [Q(A) Q(f)] = 2, 5G = X, and Q(M) = Q(X) THEOREM 2 Let G be a p-group and X an irreducible complex character of G for which the Schur index mQ(X) = 2 Then G contains a generalized quaternion section More specifically there is an irreducible character f on a subgroup K of G Received by the editors August 20, 1986 Presented at the San Antonio Meeting of the AMS on January 22, 1987 1980 Mathematics Subject Classification (1985 Revision) Primary 20C15
17 citations
TL;DR: In this article, the image of the theta series of genus n of the matrix F under the action of an irregular Hecke operator with index p was obtained in the form of a linear combination of theta-series.
Abstract: Let F be a positive-definite integral even matrix of even order. For an arbitrary prime number p and natural n one obtains an explicit expression for the image of the theta series of genus n of the matrix F, under the action of an irregular Hecke operator with index p, in the form of a linear combination of theta series.
7 citations
1 Jan 1987
TL;DR: In this paper, the authors show that the results of Lakatos' proofs and refutations have little or nothing to do with evolutionary epistemology, and the scheme he arrives at, is roughly this:
Abstract: Out understanding of the mathematical process has been and still is (rightly so) associated with Lakatos’ Proofs and Refutations. But at first sight, the results of his research have little or nothing to do with evolutionary epistemology. The scheme he arrives at, is roughly this:
7 citations
6 citations
TL;DR: In this article, the value distribution of λ − -invariants for imaginary abelian fields (and for a fixed prime number p ) was studied, where λ = 0.
4 citations
Patent•
2 Nov 1987
TL;DR: In this paper, the authors proposed a scheme to secure the safety of information by holding a master decoding key only by a receiver in secrecy, and holding an enciphering key by a transmitter that is an owner in secrecy.
Abstract: PURPOSE: To secure the safety of information by holding a master decoding key only by a receiver in secrecy, and holding an enciphering key only by a transmitter that is an owner in secrecy. CONSTITUTION: The safety can be maintained even when the length of a key is compressed by keeping both individual enciphering key and decoding key in secrecy from a person other than the transmitter and the receiver. In other words, in case of constituting a master key of 10 prime numbers of binary 20 bits and an individual key of 5 prime numbers out of them, no master key can be estimated until all of the remaining 5 prime numbers except for 5 prime numbers of the individual key are found. Around 2 16 prime numbers exist within the integer of binary 20 bits, and a combination in which five of them are taken goes to 2 16 C 5 , and it is impossible to calculate all of the cases judging from calculation amount point of view. Furthermore, even if all of the prime numbers are known, it is hard to know the master key judging from the calculation amount point of view without recognizing respective individual key. COPYRIGHT: (C)1989,JPO&Japio
4 citations
TL;DR: In this paper, a proof for all totally real and not necessarily abelian number fields having the property that 2rlQ1 is the exact 2-power dividing,v,(F) cp( 1) or the order of Kz(oP) was given.
TL;DR: Congruences, prime numbers, the unique factorisation theorem, continued fractions and, in fairly concise form, cryptography are learned in this combined book and computer software package.
Abstract: T H Jackson 1987 Bristol: Adam Hilger vi + 86 pp price £15 basic (IOP members' price £12), £100 network ISBN 0 85274 077 8 (text), 0 85274 078 6 (textX 4), 0 85274 079 4 (BBC disk 40 track), 0 85274 081 6 (BBC disk 80 track), 0 85274 080 8 (IBM disk) Much of the number theoretic content of a first-year undergraduate algebra course, and rather more, is covered by this combined book and computer software package. Thus we learn about congruences, prime numbers, the unique factorisation theorem, continued fractions and, in fairly concise form, cryptography.
TL;DR: In this article, the linear independence of the theta series arising from the classes of even positive definite ternary forms of discriminant 2p2 which have a nontrivial automorphism group was established.
Patent•
3 Jun 1987
TL;DR: A slide rule for proving Goldbach conjecture, verifying chen theorem and other topics connected with the theorem and founding a data base relating to above theory is presented in this paper, which is composed of a fundamental rule and a folding rule.
Abstract: The purpose of this invention is to make a special computational tool for studying number theory, teaching mathematics and popularizing science. This slide rule is helpful for proving Goldbach conjecture, verifying chen theorem and other topics connected with the theorem and founding a data base relating to above theory. It is composed of a fundamental rule and a folding rule. On the frontface of the fundamental rule there are an axis of the natural number, an axis of the prime number and/or an axis of the complex prime number. On the back of the folding rule there are an axis of the prime number and/or an axis of the complex prime number but on the front of the folding rule there is an axis of the natural number.
TL;DR: A set of cyclically orthogonal sequences taking real multilevel values and having different waveforms, aiming at the application to the spread-spectrum multiplex communication is derived.
Abstract: This paper derives a set of cyclically orthogonal sequences taking real multilevel values and having different waveforms, aiming at the application to the spread-spectrum multiplex communication. The real cyclic orthogonal sequence is represented by the cosine function with the phase which is an odd function, and the cross-correlation functions among sequences with the same or different lengths are discussed. By defining the convolution between sequences of the same length, a set of sequences with different waveforms and the same length can be derived. As examples of the set of sequences, the sets with length being prime number or compound number lengths are considered, which are obtained by forming convolutions of sequences with phases being monomial with odd power. Their properties are discussed. In the set of N sequences with the prime number length N, each of (N - 1) sequence takes odd multilevels between 3 and N, and the cross-correlation function between sequences is 1/√N times the sequence value. By forming the product of such sequences with different prime number lengths, a set of sequences with compound number lengths can be generated, which is a good compromise between the number of multilevels and the absolute maximum value of the cross-correlation function. N - 1 sequences of prime number lengths generated by the third-order monomial takes 3 to N multilevels and their cross-correlation function does not exceed 2/√N in the absolute value.
TL;DR: The main purpose of the present paper is to prove the following theorems, under the assumption that G^=Aut(X), the automorphism group of X; in this situation we denote by Tx(G\H1(X, QO) tne character of the natural representation of G on the first/adic cohomolqgy.fiPCX,Qz) of X, where I is any prime number different from the characteristicof k fcf.
Abstract: Introduction Let G= be a cyclic group with w = #G>l. For each divisor d of n, let %d denote the character of G of the irreducible representation over Q, the field of rational numbers, whose kernel is equal to (,od). Let k be an algebraicallyclosed ground field and X be a complete non-singular curve over k of genus Q^Z.2.The main purpose of the present paper is to prove the following theorems, under the assumption that G^=Aut(X), the automorphism group of X; in this situation we denote by Tx(G\H1(X, QO) tne character of the natural representation of G on the first/-adic cohomolqgy .fiPCX,Qz) of X, where I is any prime number differentfrom the characteristicof k fcf. Notation, 5 2 and also [41) :
TL;DR: It is shown that all these algorithms for enumerating the prime numbers up to some limit N emerge from the common idea of a sieve by the judicious use of three simple principles.
TL;DR: In this article, two methods for proving that there is a positive proportion of intervals which contain no primes and are longer than the average distance between consecutive primes are presented, and the related problem of longer than average gaps between primes is also considered.
Abstract: We present two methods for proving that there is a positive proportion of intervals which contain no primes and are longer than the average distance between consecutive primes. The first method is based on an argument of Erdos which uses a sieve upper bound for prime twins to bound the density function for gaps between primes. The second method uses known results about the first three moments for the distribution of intervals with a given number of primes. Better results are obtained by assuming that the first n moments are Poisson. The related problem of longer than average gaps between primes is also considered.
TL;DR: For those prime numbers p, for which all prime factors of p −1 are small, the two problems of finding a primitive element modulo p and of factoring univariate polynomials over finite fields of characteristic p are (deterministically) polynomial-time equivalent.