Showing papers on "Prime number published in 2009"
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TL;DR: In this article, it was shown that there are infinitely often primes differing by 16 or less in the Elliott-Halberstam conjecture and that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing.
Abstract: We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that there are infinitely often primes differing by 16 or less. Even a much weaker conjecture implies that there are infinitely often primes a bounded distance apart. Unconditionally, we prove that there exist consecutive primes which are closer than any arbitrarily small multiple of the average spacing, that is, lim inf n→∞ Pn+1-Pn/log Pn/log = 0. We will quantify this result further in a later paper.
282 citations
TL;DR: In this paper, it was shown that the unitary Cayley graph X n is hyperenergetic if and only if n has at least two prime factors greater than 2 or at least three distinct prime factors.
97 citations
TL;DR: In this article, it was shown that if G does not have any vanishing element of p-power order, then G has a normal Sylow p-subgroup and this result is a generalization of some classical theorems in Character Theory of finite groups.
58 citations
TL;DR: In this paper, it was shown that for numbers with two distinct prime factors, the bound can be improved to 6 by a generalization of the Elliott-Halberstam Conjecture.
Abstract: Let p n denote the n th prime. Goldston, Pintz, and Yildirim recently proved that li m in f (pn+1 ― p n ) n→∞ log p n = 0. We give an alternative proof of this result. We also prove some corresponding results for numbers with two prime factors. Let q n denote the n th number that is a product of exactly two distinct primes. We prove that lim inf(qn+1―q n ) n→∞ 26. If an appropriate generalization of the Elliott-Halberstam Conjecture is true, then the above bound can be improved to 6.
55 citations
Patent•
9 Dec 2009TL;DR: In this article, an elliptic curve with an order m which corresponds to a product of a first cofactor c and a prime number q is provided, wherein the order (q−1) of the multiplicative group of the prime number c corresponds to the product of another cofactor I and at least two prime divisors s 1,..., s k, wherein the at least one divisor s 1 and s k are each greater than a predetermined bound of 2 n.
Abstract: The embodiments provides a method for processing data. According to the invention, an elliptic curve with an order m which corresponds to a product of a first cofactor c and a prime number q is provided, wherein the order (q−1) of the multiplicative group of the prime number q corresponds to a product of a second cofactor I and at least two prime divisors s 1 , . . . , s k , wherein the at least two prime divisors s 1 , . . . , s k are each greater than a predetermined bound of 2 n ; and a chosen method is applied to provided data using the provided elliptic curve for providing cryptographically transformed data. The invention enables cryptographically transformed data to be provided while simultaneously minimizing the probability of a successful attack.
54 citations
TL;DR: In this article, the authors give an asymptotic formula for the number of primes p = qbn + c + a with n N, where n N is a real number such that is positive and irrational of nite type (which is true for almost all ).
Abstract: A study of certain Hamiltonian systems has led Y. Long to conjecture the existence of innitely many primes which are not of the form p = 2bn c + 1, where 1 < < 2 is a xed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately implies that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic formula for the number of primes p = qbn + c + a with n N, where ; are real numbers such that is positive and irrational of nite type (which is true for almost all ) and
52 citations
Patent•
4 Feb 2009
TL;DR: In this article, a method for performing private retrieval of information from a database is presented, in which an index corresponding to information to be retrieved from the database and a query that does not reveal the index to the database is generated.
Abstract: A method, article of manufacture and apparatus for performing private retrieval of information from a database is disclosed. In one embodiment, the method comprising obtaining an index corresponding to information to be retrieved from the database and generating a query that does not reveal the index to the database. The query is an arithmetic function of the index and a secret value, wherein the arithmetic function includes a multiplication group specified by a modulus of a random value whose order is divisible by a prime power, such that the prime power is an order of the random value. The secret value is an arithmetic function of the index that comprises a factorization into prime numbers of the modulus. The method further comprises communicating the query to the database for execution of the arithmetic function against the entirety of the database.
48 citations
TL;DR: In this paper, a generalization of the well-known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in datasets, describes with astonishing precision the statistical distribution of leading digits in the prime number sequence.
Abstract: Prime numbers seem to be distributed among the natural numbers with no law other than that of chance; however, their global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated scientists across the ages to search for local and global patterns in this distribution that could eventually shed light on the ultimate nature of primes. In this paper, we show that a generalization of the well-known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in datasets, describes with astonishing precision the statistical distribution of leading digits in the prime number sequence. Moreover, a reciprocal version of this pattern also takes place in the sequence of the non-trivial Riemann zeta zeros. We prove that the prime number theorem is, in the final analysis, responsible for these patterns.
47 citations
Book•
12 Aug 2009
TL;DR: The expanding universe of numbers has been studied extensively in the literature, see as mentioned in this paper for a character study and a discussion of the relationship between number theory and the geometry of numbers, including the number of prime numbers.
Abstract: Preface.- The Expanding Universe of Numbers.- Divisibility.- More on Divisibility.- Continued Fractions and their Uses.- Hadamard's Determinant Problem.-Hensel's P-Adic Numbers.- Notations.- Axioms.- Index.- The Arithmetic of Quadratic Forms.- The Geometry of Numbers.- The Number of Prime Numbers.- A Character Study.- Uniform Distribution and Ergodic Theory.- Elliptic Functions.- Connections with Number Theory.- Notations.- Axioms.- Index.
46 citations
Posted Content•
TL;DR: In this article, the affine sieve is applied to the case of congruence subgroups of semi-simple groups acting linearly on affine space, and the saturation number for points on such orbits at which the value of a given polynomial has few prime factors is established.
Abstract: We develop the affine sieve in the context of orbits of congruence subgroups of semi-simple groups acting linearly on affine space. In particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable and the orbit is the integers. When the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number, and thus establish the Zariski density of matrices all of whose entries are prime numbers. Among the key tools used are explicit approximations to the generalized Ramanujan conjectures for such groups and sharp and uniform counting of points on such orbits when ordered by various norms.
38 citations
TL;DR: In this article, the problem of explicit evaluation of Gauss sums in ''textsl{index 2 case}" (i.e., $G(\k^\la) (1\laN-1)$ in index 2 case with order $N$ being general even integer, where $r,N_0$ are positive integers and $N_03$ is odd).
Abstract: Let $p$ be a prime number, $q=p^f$ for some positive integer $f$, $N$ be a positive integer such that $\gcd(N,p)=1$, and let $\k$ be a primitive multiplicative character of order $N$ over finite field $\fq$. This paper studies the problem of explicit evaluation of Gauss sums in "\textsl{index 2 case}" (i.e. $f=\f{\p(N)}{2}=[\zn:\pp]$, where $\p(\cd)$ is Euler function). Firstly, the classification of the Gauss sums in index 2 case is presented. Then, the explicit evaluation of Gauss sums $G(\k^\la) (1\laN-1)$ in index 2 case with order $N$ being general even integer (i.e. $N=2^{r}\cd N_0$ where $r,N_0$ are positive integers and $N_03$ is odd.) is obtained. Thus, the problem of explicit evaluation of Gauss sums in index 2 case is completely solved.
TL;DR: In this paper, the authors count the number S(x) of quadruples for which a prime number is a determinant and satisfy the determinant condition: x ≥ 1.
Abstract: We count the number S(x) of quadruples $ {\left( {x_{1} ,x_{2} ,x_{3} ,x_{4} } \right)} \in \mathbb{Z}^{4} $
for which
$$ p = x^{2}_{1} + x^{2}_{2} + x^{2}_{3} + x^{2}_{4} \leqslant x $$
is a prime number and satisfying the determinant condition: x
1
x
4 − x
2
x
3 = 1. By means of the sieve, one shows easily the upper bound S(x) ≪ x/log x. Under a hypothesis about prime numbers, which is stronger than the Bombieri–Vinogradov theorem but is weaker than the Elliott–Halberstam conjecture, we prove that this order is correct, that is S(x) ≫ x/log x.
16 Apr 2009
TL;DR: In this paper, a probabilistic polynomial time algorithm was proposed to test whether the Jacobian of the given hyperelliptic curve of the form Y 2 = X 5 + u X 3 + v X satisfies the condition and, if so, to give the largest prime factor.
Abstract: In hyperelliptic curve cryptography, finding a suitable hyperelliptic curve is an important fundamental problem. One of necessary conditions is that the order of its Jacobian is a product of a large prime number and a small number. In the paper, we give a probabilistic polynomial time algorithm to test whether the Jacobian of the given hyperelliptic curve of the form Y 2 = X 5 + u X 3 + v X satisfies the condition and, if so, to give the largest prime factor. Our algorithm enables us to generate random curves of the form until the order of its Jacobian is almost prime in the above sense. A key idea is to obtain candidates of its zeta function over the base field from its zeta function over the extension field where the Jacobian splits.
Posted Content•
TL;DR: In this article, a necessary and sufficient condition for a complex function to be a Schur multiplier on a homogeneous tree of degree q+1 (for q between 2 and infinity) was given.
Abstract: Let X be a homogeneous tree of degree q+1 (for q between 2 and infinity) and let f be a complex function on X times X for which f(x,y) only depend on the distance between x and y in X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X times X. Moreover, we find a closed expression for the Schur norm of f. As applications, we obtain a closed expression for the completely bounded Fourier multiplier norm of the radial functions on the free (non-abelian) group on N generators (for N between 2 and infinity) and of the spherical functions on the p-adic group PGL_2(Q_q) for every prime number q.
Posted Content•
TL;DR: Using the polynomial method in additive number theory, Alon, Nathanson and Ruzsa as discussed by the authors established a new addition theorem for the set of subsums of a set satisfying $A\cap(-A)=\emptyset$ in $\mathbb{Z}/p\mathbb {Z}$: \[|Sigma(A)|\geqslant\min{p, 1+\frac{|A|(|S|+1)}{2}
Abstract: Using the polynomial method in additive number theory, this article establishes a new addition theorem for the set of subsums of a set satisfying $A\cap(-A)=\emptyset$ in $\mathbb{Z}/p\mathbb{Z}$: \[|\Sigma(A)|\geqslant\min{p,1+\frac{|A|(|A|+1)}{2}}.\]
The proof is similar in nature to Alon, Nathanson and Ruzsa's proof of the Erd\"os-Heilbronn conjecture (proved initially by Dias da Silva and Hamidoune \cite{DH}). A key point in the proof of this theorem is the evaluation of some binomial determinants that have been studied in the work of Gessel and Viennot. A generalization to the set of subsums of a sequence is derived, leading to a structural result on zero-sum free sequences. As another application, it is established that for any prime number $p$, a maximal zero-sum free set in $\mathbb{Z}/p\mathbb{Z}$ has cardinality the greatest integer $k$ such that \[\frac{k(k+1)}{2}
TL;DR: Combescure et al. as mentioned in this paper showed that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if d =pn (p a prime number and n any integer) there exists d+1 mutually unbiased bases in Cd.
Abstract: In our previous paper [Combescure, M., “Circulant matrices, Gauss sums and the mutually unbiased bases. I. The prime number case,” Cubo A Mathematical Journal (unpublished)] we have shown that the theory of circulant matrices allows to recover the result that there exists p+1 mutually unbiased bases in dimension p, p being an arbitrary prime number. Two orthonormal bases B, B′ of Cd are said mutually unbiased if ∀b∊B, ∀b′∊B′ one has that |b⋅b′|=1/d (b⋅b′ Hermitian scalar product in Cd). In this paper we show that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if d=pn (p a prime number and n any integer) there exists d+1 mutually unbiased bases in Cd. Our result relies heavily on an idea of Klimov et al. [“Geometrical approach to the discrete Wigner function,” J. Phys. A 39, 14471 (2006)]. As a subproduct we recover properties of quadratic Weil sums for p≥3, which generalizes the fact that in the prime case the quadratic Gauss sum properties fo...
TL;DR: In this paper, a smooth projective variety is defined over a number field k and two complex embeddings of k, such that the two complex manifolds induced by these embedding have non isomorphic cohomology algebras with real coefficients.
Abstract: Using constructions of Voisin, we exhibit a smooth projective variety defined over a number field k and two complex embeddings of k, such that the two complex manifolds induced by these embeddings have non isomorphic cohomology algebras with real coefficients. This contrasts with the fact that the cohomology algebras with l-adic coefficients are canonically isomorphic for any prime number l, and answers a question of Grothendieck.
Posted Content•
TL;DR: For an abelian totally real number field and an odd prime number which splits totally in the field as discussed by the authors, a functorial approach to special "$p$-units" previously built by D. Solomon using "wild" Euler systems is presented.
Abstract: For an abelian totally real number field $F$ and an odd prime number $p$ which splits totally in $F$, we present a functorial approach to special "$p$-units" previously built by D. Solomon using "wild" Euler systems. This allows us to prove a conjecture of Solomon on the annihilation of the $p$-class group of $F$ (in the particular context here), as well as related annihilation results and index formulae.
Journal Article•
TL;DR: In this article, the authors explore the subsequence of primes with prime subscripts, and derive its density and estimates for its counting function, obtaining bounds for the weighted gaps between elements of the subsequences and show that for every positive integer m there is an integer arithmetic progression with at least m of the (qn) satisfying qn = an + b.
Abstract: We explore the subsequence of primes with prime subscripts, (qn), and derive its density and estimates for its counting function. We obtain bounds for the weighted gaps between elements of the subsequence and show that for every positive integer m there is an integer arithmetic progression (an + b : n ∈ N) with at least m of the (qn) satisfying qn = an + b.
1 Jan 2009
TL;DR: In this paper, it was shown that the diophantine equation Fn = p a ± p b has only finitely many positive integer solutions (n,p,a,b), where p is a prime number and max{a, b} 2.
Abstract: In this paper, we show that the diophantine equation Fn = p a ± p b has only finitely many positive integer solutions (n,p,a,b), where p is a prime number and max{a,b} 2.
TL;DR: The author shows that the PPC is equivalent to specific boundary behavior of a function involving zeta's complex zeros, and suggests a certain hypothesis on equidistribution of prime pairs, or a speculative supplement to Montgomery's work on pair-correlation, would imply that there is an abundance ofprime pairs.
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TL;DR: In this article, the space of commuting elements in the central product of two copies of the special unitary group SU(p) is studied, where $p$ is a prime number.
Abstract: In this paper the space of commuting elements in the central product $G_{m,p}$ of $m$ copies of the special unitary group $SU(p)$ is studied, where $p$ is a prime number. In particular, a computation for the number of path connected components of these spaces is given and the geometry of the moduli space $\Rep(\mathbb Z^n, G_{m,p})$ of flat principal $G_{m,p}$--bundles over the $n$--torus is completely described for all values of $n$, $m$ and $p$.
TL;DR: Two subsets of {0,1,2,...,p-1} satisfying maxl"jj and |F"i|(modp)@?K for every 1=
TL;DR: In this paper, it was shown that all operations in the Milnor K-theory mod p of a field are spanned by divided power operations, and for all fields k 0 and all prime numbers p, all the operations K i M / p → K j M/ p commuting with field extensions over the base field k 0.
TL;DR: A construction of self-dual codes and a mass formula, which counts the number of such codes over the ring of integers modulo p3, are constructed.
Abstract: Let p be a prime number. In this paper, we consider codes over the ring $${Z_{p^3}}$$ of integers modulo p 3 and give a characterization of self-duality. This leads to a construction of self-dual codes and a mass formula, which counts the number of such codes over $${Z_{p^3}}$$ .
Patent•
16 Apr 2009TL;DR: In this article, error detection with multi-level memory cells where the number of storage levels of the memory cells is an integer power of a non-binary prime number is discussed and additional circuit and methods are disclosed.
Abstract: Memory, modules and methods for using error detection with multi-level memory cells where the number of storage levels of the memory cells is an integer power of a non-binary prime number are provided. Additional circuit and methods are disclosed.
Patent•
26 Feb 2009TL;DR: In this paper, the multiplicative inverse of the secret modulo the product of the prime numbers in each subset and, for each subset, calculates the multiplier inverses of the modulo of the product.
Abstract: A method and system distributes N shares of a secret among cooperating entities by calculating the multiplicative inverses of the secret. In one embodiment, a distributor selects N distinct prime numbers and forms unique subsets of the prime numbers, with each subset containing K of the N prime numbers (N>=K), where K is a threshold number of shares necessary to reconstruct the secret. The distributor calculates a product of the prime numbers in each subset, and, for each subset, calculates the multiplicative inverse of the secret modulo the product. A total of N shares are generated, with each share containing the multiplicative inverses and one of the prime numbers. The N shares are distributed to the cooperating entities for secret sharing.
Posted Content•
TL;DR: In this paper, the discriminant of an arbitrary elementary abelian p-extension of K was derived using Kummer theory for a finite extension of K of Qp(π) where p is a prime number and π is a primitive p-th root of 1.
Abstract: Using Kummer theory for a finite extension K of \Qp(\zeta)(where p is a prime number and \zeta a primitive p-th root of~1), we compute the ramification filtration and the discriminant of an arbitrary elementary abelian p-extension of K. We also develop the analogous Artin-Schreier theory for finite extensions of \Fp((\pi)) and derive similar results for their elementary abelian p-extensions.
TL;DR: In this article, it was shown that if l is a prime number less than 107, then for all n ≥ 1, l does not divide h n, where h n denotes the class number of ℚ(2 cos(2π/2 n+2 ).
Abstract: Let h n denote the class number of ℚ(2 cos(2π/2 n+2)). Weber proved that h n is odd for all n ≥ 1. We claim that if l is a prime number less than 107, then for all n ≥ 1, l does not divide h n .
TL;DR: Very simple sufficient conditions for the irreducibility of f(Xr) over an arbitrary unique factorization domain Z are established in this article, where the necessary and sufficient conditions are given.
Abstract: In this paper, very simple sufficient conditions for the irreducibility of f(Xr) over an arbitrary unique factorization domain Z are established.