Showing papers on "Prime number published in 1986"
TL;DR: In this paper, a probabilistic polynomial-time algorithm for computing the square root of a number x ∆ ∆ Z/P{\bf Z}, where P = 2^{S}Q + 1(Q odd, s > 0) is a prime number, is described.
Abstract: A probabilistic polynomial-time algorithm for computing the square root of a number x \in {\bf Z}/P{\bf Z} , where P = 2^{S}Q + 1(Q odd, s > 0) is a prime number, is described. In contrast to the Adleman, Manders, and Miller algorithm, this algorithm gets faster as s grows. As with the Berlekamp-Rabin algorithm, the expected running time of the algorithm is independent of x . However, the algorithm presented here is considerably faster for values of s greater than 2 .
72 citations
Book•
1 Aug 1986
TL;DR: This paper discusses a technique for multiplying numbers, modulo a prime number, using look-up tables stored in read-only memories, in the computation of number theoretic transforms implemented in a ring which is isomorphic to a direct sum of several Galois fields.
Abstract: This paper discusses a technique for multiplying numbers, modulo a prime number, using look-up tables stored in read-only memories. The application is in the computation of number theoretic transforms implemented in a ring which is isomorphic to a direct sum of several Galois fields, parallel computations being performed in each field.
68 citations
TL;DR: The number of multiplications necessary and sufficient to compute a length-2nDFT is determined and the method of derivation is shown to apply to the multiplicative complexity results of Winograd for alength-pnDFT.
Abstract: The number of multiplications necessary and sufficient to compute a length-2nDFT is determined. The method of derivation is shown to apply to the multiplicative complexity results of Winograd for a length-pnDFT, for p an odd prime number. The multiplicative complexity of the one-dimensional DFT is summarized for many possible lengths.
66 citations
Proceedings Article•
1 Jan 1986
TL;DR: A probabilistic polynomial-time algorithm for computing the square root of a number x/P Z, where P = 2 S + 1 Q + 1(Q odd, s > 0) is a prime number, is described.
Abstract: A probabilistic polynomial-time algorithm for computing the square root of a number x \in {\bf Z}/P{\bf Z} , where P = 2^{S}Q + 1(Q odd, s > 0) is a prime number, is described. In contrast to the Adleman, Manders, and Miller algorithm, this algorithm gets faster as s grows. As with the Berlekamp-Rabin algorithm, the expected running time of the algorithm is independent of x . However, the algorithm presented here is considerably faster for values of s greater than 2 .
60 citations
1 Oct 1986
TL;DR: In this article, the Gross-Koblitz formula and a formula of diamond were used to prove the congruence A=(1+((2 p-1 -1)/2)× 2a-(p/2a))(mod p 2 ) (p a prime number ≡ 1 (mod 4), p = a 2 + b 2 (a, b, ∈ Z, a≡ 1(mod 4))), proposed by F. Beukers.
Abstract: The Gross-Koblitz formula and a formula of Diamond are used to prove the congruence A=(1+((2 p-1 -1)/2)× 2a-(p/2a))(mod p 2 ) (p a prime number ≡ 1 (mod 4), p = a 2 + b 2 (a, b , ∈Z, a≡ 1 (mod 4))), proposed by F. Beukers which refines the well-known congruence A ≡ 2a (mod p) for the binomial coefficient A=((p-1)/2 (p-1)/4).
29 citations
TL;DR: In this paper, a vanishing criterion for the Friedlander-Quillen conjecture was established for the Hopf algebraic structure on the big etale site for k, and a new proof of the theorem of Friedlander Mislin which avoids characteristic 0 theory was established.
Abstract: Let k be an algebraically closed field, and let l be a prime number not equal to char(k). Let X be a locally fibrant simplicial sheaf on the big etale site for k, and let Y be a k scheme which is cohomologically proper. Then there is a Kunneth-type isomorphism which is induced by an external cup-product pairing. Reductive algebraic groups G over k are cohomologically proper, by a result of Friedlander and Parshall. The resulting Hopf algebra structure on may be used together with the Lang isomorphism to give a new proof of the theorem of Friedlander-Mislin which avoids characteristic 0 theory. A vanishing criterion is established for the Friedlander-Quillen conjecture.
12 citations
TL;DR: In this article, the authors discuss the nilpotency index of the radical of group algebra IX, and show that for a group G of p-length 1, t (G) = t (P) for any p-Sylow subgroup, and that t(G) is the inverse of t(P) in the radical J (KG) of the group algebra K of G over K.
Abstract: Publisher Summary This chapter discusses the nilpotency index of the radical of group algebra IX. Let p be a fixed prime number, G be a finite p-solvable group with a p-Sylow subgroup P, K be a field of characteristic p, and t (G) be the nilpotency index of the radical J (KG) of group algebra K of G over K. It follows from Morita's theorem and Villamayor's theorem that t (G) = t (P) for a group G of p-length 1.
12 citations
TL;DR: An outline is given to show how previously developed techniques can be applied to improve the efficiency of the algorithm to O(n/loglogn) time and space.
Abstract: A new algorithm is presented for finding all primes between 2 and an incrementally increasing value n. The algorithm executes in linear arithmetic time and space. An outline is given to show how previously developed techniques can be applied to improve the efficiency of the algorithm to O(n/loglogn) time and space.
12 citations
TL;DR: It is shown, when P is a Mersenne prime, implementation of this algorithm on a processor, designed especially for mod P arithmetic operations, produces a more efficient algorithm with respect to the amount of program statements and number of operations.
Abstract: The extended Euclidean algorithm is typically used to calculate multiplicative inverses over finite fields and rings of integers. The algorithm presented here has approximately the same number of average iterations and maximum number of iterations. It is shown, when P is a Mersenne prime, implementation of this algorithm on a processor, designed especially for mod P arithmetic operations, produces a more efficient algorithm with respect to the amount of program statements and number of operations. It is then shown heuristically, when the division and multiplications are performed simultaneously, the Euclidean algorithm has fewer subiterations.
10 citations
1 Apr 1986
TL;DR: In this paper, strong approximation results for the residual properties of free groups have been shown to be strong in the sense that the intersection of all kernels of Op, for an infinite set of primes, is trivial and so F is residually-{SL(3, p) I p prime} if the center of F is trivial.
Abstract: By using "strong approximation" results for linear group a question of Magnus on residual properties of free groups is answered affirmatively. Let W be a nonempty class of groups and F a group. F is said to be residually-' if for every g # 1 in IF there is an epimorphism 0: G -* C where C is a group belonging to W s.t. p(g) # 1. Equivalently the intersection of all normal subgroups N of IF for which F/N belongs to W is the unit element of G. The residual properties of the free groups F,n received a lot of attention: They had been known to be residually finite even before P. Hall coined the phrase "residually finite". They are also known to be residually-W(p) where W(p) is the class of finite p-groups for a fixed prime p. In 1968 Wilhelm Magnus surveyed the state of the art of residually finite groups and wrote [M, p. 309]: "For n > 2, Fn is not only residually-W(p) for every prime number p, it is also residually-A and residually-PSL(2, pk) for fixed k and variable p, where A denotes the class of alternating groups.... Here, incidentally the story ends. It is not even known whether F2 is residually-PSL(3, p) where p runs though all primes or residually-PSL(2, 2k) where k runs through the positive integers." In this note we observe that these two questions can be answered affirmatively by using recent deep "strong approximation" theorems for linear groups proved by Weisfeiler [W] (see also Nori [N] and Matthews-Vaserstein-Weisfeiler [MVW]). These results say, for example, that if F is a subgroup of SL(3, Z) which is Zariski dense in SL(3, C), then its closure in the congruence topology of SL(3, Z) is of finite index in SL(3, Z). In particular, for almost all primes p, F is mapped onto SL(3, p) under the canonical map op: SL(3, Z) -> SL(3, Z/p Z). The intersection of all kernels of Op, for an infinite set of primes, is trivial and so F is residually-{SL(3, p) I p prime} (and also residually-PSL(3, p) if the center of F is trivial). To answer Magnus' first question it remains now to check that F2 can be embedded as a Zariski-dense subgroup in SL(3, Z). Indeed, SL(3, Z) is Zariski-dense in SL(3, C) by Borel Density Received by the editors July 19, 1985 and, in revised form, November 1, 1985. 1980 Mathematics Subject Classification (1985 Revision). Primary 20E26, 20H05. ' This work was done while the author was visiting the University of Chicago, whose warm hospitality and support are gratefully acknowledged. ?01986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page
9 citations
TL;DR: In this paper, the author obtained estimates for multiple trigonometric sums with a general polynomial in the exponent whose variables of summation take the values of the successive prime numbers.
Abstract: The author obtains estimates for multiple trigonometric sums with a general polynomial in the exponent whose variables of summation take the values of the successive prime numbers. The precision of the estimates is similar to that of the estimates in the paper "Multiple trigonometric sums and their applications" by Arkhipov, Karatsuba and Chubarikov (Math. USSR Izv. (1980), 1–54). Bibliography: 12 titles.
TL;DR: In this article, it was shown that if R satisfies the isomorphism theorem, then it is a nicely decomposing ring in the following sense: inv(R) denotes the set of prime numbers that invert in R.
Abstract: Publisher Summary This chapter describes a conjecture concerning the isomorphism problem for commutative group algebras. Suppose R is a commutative ring with identity, and let G and H be abelian groups (possibly infinite). Let GR (respectively HR) be the direct sum of the p-components of torsion in g (respectively h) such that the prime p inverts in R. It was shown earlier that if R satisfies the isomorphism theorem, then it is a nicely decomposing ring in the following sense: Let inv(R) denote the set of prime numbers that invert in R. Suppose R is a finite product of indecomposable rings, each of characteristic O. Then, R satisfies the isomorphism theorem only if R is an ND-ring.
Journal Article•
TL;DR: In this paper, it was shown that there exists a prime number (precisely an infinite prime number) p such that none of the α 1, α 2, α n natural numbers α n is a primitive root of p.
Abstract: In this paper, we give an interesting result that, given n natural numbers α1α2…αn, there exists a prime number (precisely an infinite prime number) p such that none of the α1, α2,…, αn is a primitive root of p. Finally, we have got a result about a lower bound of g(p)- the least positive primitive root of p, just similar to Turan's, but our proof is much simpler than his.
TL;DR: The analysis of the classification of the finite groups according to the number of conjugacy classes has been studied in this paper, where new properties of the numbersr� G(xN) = |{Cl� G isEnabled (g)|Cl� G� (g)∩xN ≠ O} (whereG is a finite group andN is a normal subgroup ofG) were obtained.
Abstract: We get new properties of the numbersr
G(xN) = |{Cl
G
(g)|Cl
G
(g)∩xN ≠ O} (whereG is a finite group andN is a normal subgroup ofG) that are useful in the analysis of the classification of the finite groups according to the number of conjugacy classes.
TL;DR: In this article, an analogue of Vinogradov's uniform distribution result for prime numbers in the context of hyperbolic flows and their closed orbits is studied. And the authors obtain estimates for the Hausdorff dimension of certain exceptional sets.
Abstract: In this note we study an analogue of Vinogradov’s uniform distribution result for prime numbers in the context of hyperbolic flows and their closed orbits. We obtain estimates for the Hausdorff dimension of certain exceptional sets.
Patent•
11 Dec 1986
TL;DR: In this article, the authors proposed to shorten the prime number deciding time by deciding successively whether a positive integer is a prime number or not, by using the first -the fourth deciding means and a random number generating means.
Abstract: PURPOSE:To decrease a calculation processing and to shorten the prime number deciding time by deciding successively whether a positive integer is a prime number or not, by using the first - the fourth deciding means and a random number generating means. CONSTITUTION:A given positive integer N is inputted to the first deciding means 9, and unless it is a prime to each other to a positive integer M, it is decided that N is not a prime number. In this case, when it is decided that N is not a prime number, if a remainder which is dividedly A N N with respect to an integer A generated by a random number generating means 10 is not '1' nor N-1, the second deciding means 11 decides that N is not a prime number. When this remainder is different from an integer which is determined by depending on N and A, the third deciding means decides that N is not a prime number. Also, when both the second deciding means 11 and the third deciding means 12 does not decided that N is not a prime number with respect to an optional number of random numbers generated by the random number generating means 10, the fourth deciding means 13 decides that N is a prime number.
1 Jan 1986
TL;DR: In this article, the problem of approximating the number theoretic function π(x) with respect to all prime numbers p ≤ x by means of the function x/log x was studied.
Abstract: Until now the approximation of individual real numbers has been studied with the help of integers, and for this the methods of Analysis and Geometry have been seen to be helpful. From now on the reverse problem will be attacked: number theoretic functions, the calculation of which for large values is severely limited because of the irregularity of their behaviour, will be represented approximately by known functions from differential and integral calculus with easily described behaviour. Moreover the methods of analysis are also available for these approximation problems. For example in Chapter 5 we prove the prime number theorem, which has for its subject the approximation of the number theoretic function π(x), which counts all prime numbers p ≤ x by means of the function x/log x. In this chapter we develop the notation and necessary techniques, discuss how good an approximation is, and present some simple examples.
Journal Article•
TL;DR: In this paper, the set of questions equivalent and similar to those of twin numbers is considered and a theorem of relative density is included as ewll an inductive process of functional generation of the prime numbers.
Abstract: This paper considers questions equivalent and similar to those of twin numbers. Two prime numbers are said to be twin if they differ in two units. Except for the pair (3,5), they may be expressed as ($6n-1,6n+1$). The set of the multiples of 6 which separate twin numbers is studied. A theorem of relative density is included ($\S$ 10) as ewll an inductiv process of functional generation of the prime numbers ($\S$ 11).
TL;DR: In this article, the parity of Hn(a, b, c) is determined in all cases and further congruences are given modulo 4 or 16 when Hn is even or if n + 2 is a prime number greater than 3.
TL;DR: In this paper, Ma-Tu et al. used discriminator theory to determine equations which have as spectrum the closure under multiplication of the spectrum of a given first-order sentence.
Abstract: A sentence in the language L for the empty type (so the unique relation symbol is equality) must express only properties concerning the cardinality of structures. A sentence Q in L is said to exclude a finite set S of positive integers when the cardinalities of the finite models of Q are exactly the positive integers in the complement S of ~r When S is closed under multiplication and contains 1, such a sentence has to be logically equivalent to a strict Horn sentence [Gr 1979]. K. I. Appel [Ap 1959] determined for any S as above a strict Horn V::l-sentence in L excluding ~r Its size grows superexponentially with the maximum m of S. In the paper [Tu 1984] a polynomial-sized sentence Hp excluding S was found in the case that S consists of a single prime number p. In fact, Hp has size O(p 5 logp). Later also a solution of size O(m m log m), when rn is the maximum of ~r to the general problem was given [Ma-Tu 1985]. This general solution uses discriminator theory to determine equations which have as spectrum the closure under multiplication of the spectrum of a given first order sentence. This useful technique goes back to R. McKenzie [McK 1975]. The particular solution Hp, which excludes the prime p, was worked out [Tu 1984] by using abelian groups instead of discriminator algebras. Here, using the simpler structure of a set with a permutation, we are able to reduce the size of Horn sentences excluding a prime p to O(p 3 log p). We consider the variables {xi:O<-i
1 Jan 1986
TL;DR: In this paper, a comparative etude entre la theory de corde unifiee des quatre interactions fondamentales and quatre fonctions zeta is presented.
Abstract: On presente une etude comparative entre la theorie de corde unifiee des quatre interactions fondamentales et quatre fonctions zeta
TL;DR: In this article, it was shown that up to conjugacy, there exist exactly 4 soluble minimal irreducible subgroups in,,, and, where each is a Sylow 2-subgroup of and,, and are minimal transitive groups of permutation matrices of degree and are metabelian groups of class 3 with three generators.
Abstract: The author describes the soluble minimal irreducible subgroups of , where and are prime numbers, q$ SRC=http://ej.iop.org/images/0025-5734/55/1/A03/tex_sm_2990_img4.gif/>, , and is an arbitrary subfield of the field of real numbers. He proves that up to conjugacy, there exist exactly 4 soluble minimal irreducible subgroups in , , , and , where each is a Sylow 2-subgroup of and , , and are minimal transitive groups of permutation matrices of degree and are metabelian groups, each of which is generated by two matrices, and and are soluble groups of class 3 with three generators: where is the order of the number 2 modulo , is the order of modulo , and is the order of modulo .The properties of subspaces generated by the rows of circulants over a prime finite field are investigated. The connection between these properties and the problem of describing certain classes of minimal irreducible linear groups is indicated.Bibliography: 18 titles.
TL;DR: In this article, the Gross-Koblitz formula and a formula of diamond are used to prove the congruence A = 1+ 2 p−1 −1 2 2a− p 2a (mod p 2 ) (p a prime number ≡ 1 (mod 4), p = a2 + b2 (a, b ∈ Z, a ≡ 1(mod 4))), proposed by F. Beukers which refines the well-known congruences A ≡ 2a(mod p) for the binomial coefficient A= p− 1 2
1 Nov 1986
TL;DR: A new probabilistie primality test is presented, different from the tests of Miller, Solovay-Strassen, and Rabin in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption.
Abstract: This paper presents a new probabilistie primality test. Upon termination the test outputs "composite" or "prime", along with a short proof of correctness, which can be verified in deterministic polynomial time. The test is different from the tests of Miller [M], Solovay-Strassen [SSI, and Rabin [R] in that its assertions of primality are certain, rather than being correct with high probability or dependent on an unproven assumption. Thc test terminates in expected polynomial time on all but at most an exponentially vanishing fraction of the inputs of length k, for every k. This result implies: • There exist an infinite set of primes which can be recognized in expected polynomial time. • Large certified primes can be generated in expected polynomial time. Under a very plausible condition on the distribution of primes in "small" intervals, the proposed algorithm can be shown ' to run in expected polynomial t ime on eve ry i n p u t . This * R e s e a r c h s u p p o r t e d i n p a r t by N S F G r a n t 8 5 0 9 9 0 5 D C R Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, a n d notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. © 1986 ACM 0-89791-193-8/86/0500/0316 $00.75 316 condition is implied by Cramer 's conjecture. The methods employed are from the theory of elliptic curves over finite fields. 1. I N T R O D U C T I O N I . I T e s t i n g P r i m a l i t y : B r i e f R e v i e w Distinguishing prime numbers from composites has intrigued mathemat ic ians as early as about 274 B.C, when the sieve algorithm of Eratosthenes has been allegedly recorded. Much progress has been made on this problem since tile 17th century by Fermat, Euler, Legendre and Gauss. With the arrival of fast computat ional devices new algorithmic ideas based on the work of Fermat and Gauss were proposed and implemented (see [D],[BLS]). These algorithms mostly relied on factoring and thus where impractical for even moderate size inputs. The interest in primality in complexity theory was invoked by the exciting primality tests of Miller[M], Solovay and Srassen [SS], and Rabin [R]. Miller's algorithm [M] is a deterministic polynomial t ime procedure, which when answering "composite" gives a proof of correctness, and when answering "prime" does not. The assertions of primality made by the algori thm are always correct if if the Extended Riemann Hypothesis (ERH) is true. However, if the ERII is false, the numbers declared prime may still be composite. Thus, tile ERII is not used to bound the running time of the algorithm, but to vouch for the correctness of the answer. The probabilistic primality tests of SolovayStrassen [SS] and Rabin [SS], essentially perform a probabilistic search for a proof of compositeness. The failure of this search, provides circumstantial evidence that the number is not composite. These algorithms always terminate in polynomial time on every input. Upon termination they declare the input either composite or probably prime. When a number is declared "composite", a short (verifiable in deterministic polynomial time) proof (certificate) of compositeness is provided. When a number is declared "probably prime", then it is a prime with very high probability, but no certainty is provided. The fastest deterministic algorithm known is due to Adleman, Pomerance and Rumley [APR] (followed by Choen-Lenstra[CL]) and runs in time O(kc log los k) on inputs of length k. The answers of this algorithm are always correct. Unfortunately, it is not only slow but, like its predecessors, does do not provide us with a short certificate (i.e polynomial time verifiable proof) of its assertions of primality. As discussed above, finding a short certificate of compositeness can be done quickly probabilistically. But, how about short proofs of primality? Although it is not as obvious as in the case of compositeness, Pra t t [P] has shown that short proofs of primality do exist (i.e the set PRIMES is in NP). Unfortunately, finding a Pratt-certificate for a given prime involves being able to factor quickly, which is hard. Partial progress toward finding short proof of primality quickly was made by Furer IF]. He shows a Las Vegas (always correct, probably fast) algorithm distinguishing between n a product of two primes and the n a prime (provided n ~ 1 mod 24). To summarize, the following questions remain open: • Is there an infinite set of primes which can be recognized in expected polynomial time ? • Can random large certified primes be generated in expected polynomial time? • Is there a probabilistic primality test which is alaways correct and probably fast on every prime input, i.e a Las Vegas primality test ? 1.2 O u r Resul t s In this paper, we propose a probabilistic algorithm which upon termination outputs either "prime" or "composite", along with a short proof (certificate) of correctness. The proof of correctness can be verified by a deterministic polynomial Lime algorithm. We prove the following. T h e o r e m 1: Given any prime p of length k, our algorithm outputs a certificate of correctness of size O(k2), which can be verified correct in O(k 4) deterministic time. T h e o r e m 3: For every size k ~> 0, our algorithm terminates in expected polynomial time on at least 1 0(2 ' ~ ' ~ ' ~ ) of the prime inputs of length k. Note that the fraction of primes for which we could not prove that the algorithm terminates in expected polynomial time is smaller than any polynomial in k fraction. Let ~(x) denote the number of primes smaller than z. Theorem 2: Our algorithm terminates in expected polynomial time on every input if the following conjecture is true:
TL;DR: In this article, the authors give upper and lower bounds for a positive integer x in terms of the greatest square-free divisor of x y z for positive integers such that x = y+z and ged (x,y,z)=1.
Abstract: Letx, y andz be positive integers such thatx=y+z and ged (x,y,z)=1. We give upper and lower bounds forx in terms of the greatest squarefree divisor ofx y z.