Showing papers on "Prime number published in 2003"
TL;DR: An algorithm that computes the prime numbers up to N using O(N/log log N) additions and N 1/2+o(1) bits of memory is introduced.
Abstract: We introduce an algorithm that computes the prime numbers up to N using O(N/log log N) additions and N 1/2+o(1) bits of memory. Tie algorithm enumerates representations of integers by certain binary quadratic forms. We present implementation results for this algorithm and one of the best previous algorithms.
117 citations
Book•
1 Jan 2003TL;DR: The prime number theorem as mentioned in this paper is one of the great classical theorems of mathematics, and it can be used to obtain a simple formula that tells us how many primes we can expect to find that are less than any integer we might choose.
Abstract: At first glance the prime numbers appear to be distributed in a very irregular way amongst the integers, but it is possible to produce a simple formula that tells us (in an approximate but well defined sense) how many primes we can expect to find that are less than any integer we might choose. The prime number theorem tells us what this formula is and it is indisputably one of the great classical theorems of mathematics. This textbook gives an introduction to the prime number theorem suitable for advanced undergraduates and beginning graduate students. The author's aim is to show the reader how the tools of analysis can be used in number theory to attack a 'real' problem, and it is based on his own experiences of teaching this material.
78 citations
TL;DR: In this paper, it was shown that every interval [x(1−Δ−1),x] contains a prime number with Δ = 28 314 000 and provided x⩾10 726 905 041.
60 citations
TL;DR: A short review of Schrodinger Hamiltonians for which the spectral problem has been related in the literature to the distribution of the prime numbers is presented in this paper, where a possible connection between prime numbers and centrifugal inversions in black holes is discussed.
Abstract: A short review of Schrodinger Hamiltonians for which the spectral problem has been related in the literature to the distribution of the prime numbers is presented here. We notice a possible connection between prime numbers and centrifugal inversions in black holes and suggest that this remarkable link could be directly studied within trapped Bose–Einstein condensates. In addition, when referring to the factorizing operators of Pitkanen and Castro and collaborators, we perform a mathematical extension allowing a more standard supersymmetric approach.
41 citations
14 Aug 2003
TL;DR: An extremely fast point counting algorithm for hyperelliptic curves of type y 2=x 5+ax over given large prime fields \(\mathbb{F}_{p}\), e.g. 80-bit fields is proposed.
Abstract: Counting rational points on Jacobian varieties of hyperelliptic curves over finite fields is very important for constructing hyperelliptic curve cryptosystems (HCC), but known algorithms for general curves over given large prime fields need very long running time. In this article, we propose an extremely fast point counting algorithm for hyperelliptic curves of type y 2=x 5+ax over given large prime fields \(\mathbb{F}_{p}\), e.g. 80-bit fields. For these curves, we also determine the necessary condition to be suitable for HCC, that is, to satisfy that the order of the Jacobian group is of the form l· c where l is a prime number greater than about 2160 and c is a very small integer. We show some examples of suitable curves for HCC obtained by using our algorithm. We also treat curves of type y 2=x 5+a where a is not square in \(\mathbb{F}_{p}\).
39 citations
TL;DR: In this article, a short review of Schroedinger hamiltonians for which the spectral problem has been related in the literature to the distribution of the prime numbers is presented, and it is suggested that this remarkable link could be directly studied within trapped Bose-Einstein condensates.
Abstract: A short review of Schroedinger hamiltonians for which the spectral problem has been related in the literature to the distribution of the prime numbers is presented here. We notice a possible connection between prime numbers and centrifugal inversions in black holes and suggest that this remarkable link could be directly studied within trapped Bose-Einstein condensates. In addition, when referring to the factorizing operators of Pitkanen and Castro and collaborators, we perform a mathematical extension allowing a more standard supersymmetric approach
35 citations
TL;DR: Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed and it is established that both inverted techniques, when applied to the same set of levels, lead to theSame fractal potential.
Abstract: Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D=1.5 for the Riemann zeros and D=1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbor spacings of the Riemann zeros are known to be chaotically distributed, whereas the primes obey almost Poissonlike statistics. Our findings show that the fractal dimension is dependent on both level statistics and spectral rigidity, Delta(3), of the energy levels.
35 citations
TL;DR: A number of discriminants for which C(Δ) is larger than any previously known examples are found, under the assumption of the Extended Riemann Hypothesis.
Abstract: Hardy and Littlewood's Conjecture F implies that the asymptotic density of prime values of the polynomials fA(x) = x2 + x + A, A ∈ Z, is related to the discriminant Δ = 1 - 4A of fA(x) via a quantity C(Δ). The larger C(Δ) is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant Δ. A technique of Bach allows one to estimate C(Δ) accurately for any Δ 0 given the class number and regulator of the real quadratic order with discriminant Δ. The Manitoba Scalable Sieve Unit (MSSU) has shown us how to rapidly generate many discriminants Δ for which C(Δ) is potentially large, and new methods for evaluating class numbers and regulators of quadratic orders allow us co compute accurate estimates of C(Δ) efficiently, even for values of Δ with as many as 70 decimal digits. Using these methods, we were able to find a number of discriminants for which, under the assumption of the Extended Riemann Hypothesis, C(Δ) is larger than any previously known examples.
27 citations
TL;DR: For any integer n > 1, if |an| = 2n − 1, then there are values of a0, a1,..., an with a0 = ± 1 such that the polynomial f(x) in (1) is reducible as discussed by the authors.
Abstract: is irreducible over the rationals. I. Schur (in [10]) obtained this result in the special case that an = ±1 and used it to establish the irreducibility of H2n(x) where Hm(x) is the mth Hermite polynomial. The result stated above is best possible in the sense that, for any integer n > 1, if |an| = 2n − 1, then there are values of a0, a1, . . . , an with a0 = ±1 such that the polynomial f(x) in (1) is reducible. Indeed, if |an| = 2n − 1 and a0 = ±1, then one can take an−2 = an−3 = · · · = a1 = 0 and an−1 to be one of the four numbers ±u2n−2 ± 1 to deduce that f(x) is divisible by x2 − 1 (or, if desired, by x2 + 1). There are other examples of reducibility that can occur when |an| = 2n−1. The polynomial f(x) defined by u12f(x) = 11x12 + 1188x8 + 6930x4 + 10395 = 11(x4 + 3)(x8 + 105x4 + 315)
27 citations
12 Aug 2003
TL;DR: EQFT is presented, which is related to the Miller-Rabin test and the Quadratic Frobenius test by Grantham, and upper bounds for the error in case a prime is sought by incremental search from a random starting point are obtained.
Abstract: We present an Extended Quadratic Frobenius Primality Test (EQFT), which is related to the Miller-Rabin test and the Quadratic Frobenius test (QFT) by Grantham. EQFT takes time about equivalent to 2 Miller-Rabin tests, but has much smaller error probability, namely 256/331776^t for t iterations of the test in the worst case. EQFT extends QFT by verifying additional algebraic properties related to the existence of elements of order dividing 24. We also give bounds on the average-case behaviour of the test: consider the algorithm that repeatedly chooses random odd k bit numbers, subjects them to t iterations of our test and outputs the first one found that passes all tests. We obtain numeric upper bounds for the error probability of this algorithm as well as a general closed expression bounding the error. For instance, it is at most 2^{-143} for k=500, t = 2. Compared to earlier similar results for the Miller-Rabin test, the results indicate that our test in the average case has the effect of 9 Miller-Rabin tests, while only taking time equivalent to about 2 such tests. We also give bounds for the error in case a prime is sought by incremental search from a random starting point.
27 citations
TL;DR: A list of primitive permutation groups of square-free degree is given in this article, where all such groups are either solvable and act on a prime number of points, or are almost simple.
Abstract: The paper gives lists of all the primitive permutation groups of squarefree degree. All such groups are either solvable and act on a prime number of points, or are almost simple. Among the almost simple examples, the groups of Lie type have rank at most 2, or the point stabilizer is a parabolic subgroup. 2000 Mathematics Subject Classification 20B15.
TL;DR: All {δ(q + 1), δ 3, q}-minihypers, δ small, q = ph0, h ≥ 1, for a prime number p0 ≥ 7, which arise from a maximal partial spread of deficiency δ are classed.
Abstract: This article classifies all lδ(q + 1), δs 3, qr-minihypers, δ small, q e ph0, h ≥ 1, for a prime number p0 ≥ 7, which arise from a maximal partial spread of deficiency δ When q is a third power, the minihyper is the disjoint union of projected PG(5, \sqrt[3]{q})'ss when q is a square, also Baer subgeometries PG(3, \sqrt{q}) can occur This leads to a discrete spectrum for the small values of the deficiency δ of the corresponding maximal partial spreads
TL;DR: A modification of Adleman and Huang's method which runs conjecturally in expected time Lp, which is conjectured to compute a logarithm in a prime field whose cardinality p is of the form $r^e-s$, with r and s small in absolute value.
Abstract: Let p be a prime number and n a positive integer, and let q=pn. Adleman and Huang [Inform. and Comput., 151 (1999), pp. 5--16] have described a version of the function field sieve which is conjectured to compute a logarithm in the field of q elements in expected time Lq[1/3;(32/9)1/3+o(1)], where Lq[s;c]=exp(c(log q)s(log log q)1-s) and the o(1) is for $q\to\infty$ under the constraint that p6\leq n$. In this paper, we present a modification of their method which runs conjecturally in expected time Lq[1/3;(32/9)1/3+o(1)] so long as $q\to\infty$ with $p\leq n^{o(\sqrt{n})}$. The technique we use can also be applied to the special number field sieve and results in an algorithm which, in expected time Lp[1/3;(32/9)1/3+o(1)], is conjectured to compute a logarithm in a prime field whose cardinality p is of the form $r^e-s$, with r and s small in absolute value.
15 Jun 2003
TL;DR: A new multiplication algorithm for the implementation of elliptic curve cryptography (ECC) over the finite extension fields GF(p/sup k/) where p is a prime number greater than 2k is presented.
Abstract: We present a new multiplication algorithm for the implementation of elliptic curve cryptography (ECC) over the finite extension fields GF(p/sup k/) where p is a prime number greater than 2k. In the context of ECC we can assume that p is a 7-to-10-bit number, and easily find values for k which satisfy: p>2k, and for security reasons log/sub 2/(p)/spl times/k/spl sime/160. All the computations are performed within an alternate polynomial representation of the field elements which is directly obtained from the inputs. No conversion step is needed. We describe our algorithm in terms of matrix operations and point out some properties of the matrices that can be used to improve the design. The proposed algorithm is highly parallelizable and seems well adapted to hardware implementation of elliptic curve cryptosystems.
TL;DR: In this paper, the problem of upper bound estimation for the least quadratic non-residue modulo prime number on special arithmetic sequences such as f(n) = [αn] and f(nc] was considered.
Abstract: In this paper we consider the problem of an upper bound estimate for the least quadratic non-residue modulo prime number on special arithmetic sequences such as f(n) = [αn] and f(n) = [nc].
TL;DR: It is proved that if βj ≡ 1 (mod 3) orβj ≡ 2 (mod 5) for all j, 1 ≤ j ≤ k, then 3|n is perfect, where σ(n) denotes the sum of the positive divisors of n.
Abstract: We say n ∈ N is perfect if σ(n) = 2n, where σ(n) denotes the sum of the positive divisors of n. No odd perfect numbers are known, but it is well known that if such a number exists, it must have prime factorization of the form n = pα Φj=1k qj2βj, where p,q1,..., qk are distinct primes and p ≡ α ≡ 1 (mod 4). We prove that if βj ≡ 1 (mod 3) or βj ≡ 2 (mod 5) for all j, 1 ≤ j ≤ k, then 3|n. We also prove as our main result that Ω(n) ≥ 37, where Ω(n) = α + 2 Σj=1k βj. This improves a result of Sayers (Ω(n) ≥ 29) given in 1986.
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TL;DR: In this paper, it was shown that any complex Hopf algebra of dimension pq is semisimple and hence isomorphic to either a group algebra or to the dual of a group algebra.
Abstract: In this paper we contribute to the classification of Hopf algebras of dimension pq, where p,q are distinct prime numbers. More precisely, we prove that if p and q are odd primes with p
18 May 2003
TL;DR: This work considers the requirements for good uniform random number generators, and describes a class of generators whose period is a Mersenne prime or a small multiple of a MERSenne prime.
Abstract: Pseudo-random numbers with long periods and good statistical properties are often required for applications in computational finance. We consider the requirements for good uniform random number generators, and describe a class of generators whose period is a Mersenne prime or a small multiple of a Mersenne prime. These generators are based on "almost primitive" trinomials, that is trinomials having a large primitive factor. They enable very fast vector/parallel implementations with excellent statistical properties.
11 Mar 2003
TL;DR: In this paper, it was shown that if H is a subgroup of a p-solvable group G, then v p(H) divides v p (G) and vp(G) is the number of Sylow p-subgroups.
Abstract: If G is a finite group and p is a prime number, let vp(G) be the number of Sylow p-subgroups of G. If H is a subgroup of a p-solvable group G, we prove that v p (H) divides v p (G).
TL;DR: A test with a single seed depending only on h, in contrast with the unmodified test, which, as proved by W. Bosma in Explicit primality criteria for h 2 k ± 1, Math.
Abstract: Let {s k , k > 0} be the sequence defined from a given initial value, the seed, so, by the recurrence s k+1 = s 2 k - 2,k > 0. Then, for a suitable seed so, the number M h,n = h.2 n - 1 (where h < 2 n is odd) is prime iff s n-2 ≡ 0 mod M h,n . In general so depends both on h and on n. We describe a slight modification of this test which determines primality of numbers h.2 n ±1 with a seed which depends only on h, provided h ≢ 0 mod 5. In particular, when h = 4 m - 1, m odd, we have a test with a single seed depending only on h, in contrast with the unmodified test, which, as proved by W. Bosma in Explicit primality criteria for h 2 k ± 1, Math. Comp. 61 (1993), 97-109, needs infinitely many seeds. The proof of validity uses biquadratic reciprocity.
1 Oct 2003
TL;DR: A polynomial time algorithm is given for computing the Igusa local zeta function attached to aPolynomial in one variable, with splitting field $\QTR{Bbb}{Q}$, and a prime number $p$.
Abstract: We give a polynomial time algorithm for computing the Igusa local zeta function Z(s;f) attached to a polynomial f(x)2 Z[x], in one variable, with splitting fleld Q, and a prime number p. We also propose a new class of linear feedback shift registers based on the computation of Igusa’s local zeta function.
TL;DR: MoreMoree and Stewart as discussed by the authors gave a lower bound for the number of S-units in a number field of degree n composed of prime ideals which lie outside a given finite set of positive rational numbers T and which have norm Y.
Abstract: Among other things we show that for each n-tuple of positive rational numbers (a_1,..., a_n) there are sets of primes S of arbitrarily large cardinality s such that the solutions of the equation a_1x_1+..+a_nx_n=1 with x_1,..,x_n S-units are notcontained in fewer than exp((4+o(1)) s^{1/2} (log s)^{-1/2} proper linear subspaces of C^n. This generalizes a result of Erdos, Stewart and Tijdeman for S-unit equations in two variables. Further, we prove that for any algebraic number field K of degree n, any integer m with 1<= m
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TL;DR: In this article, the statistical properties of the distances and their increments for a sequence comprising the first $5\times 10^7$ prime numbers were studied and the histogram of the increments followed an exponential distribution with superposed periodic behavior of period three, similar to previously reported period six oscillations for the distances.
Abstract: The difference between two consecutive prime numbers is called the distance between the primes. We study the statistical properties of the distances and their increments (the difference between two consecutive distances) for a sequence comprising the first $5\times 10^7$ prime numbers. We find that the histogram of the increments follows an exponential distribution with superposed periodic behavior of period three, similar to previously-reported period six oscillations for the distances.
TL;DR: In this paper, it was shown that every sufficiently large even integer can be written as a sum of a Piatetski-Shapiro prime and an almost-prime.
Abstract: Suppose that . We prove a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes p = [n
1/γ and show that every sufficiently large even integer can be written as a sum of a Piatetski-Shapiro prime and an almost-prime.
TL;DR: The main idea for finding these Carmichael numbers is that the first author's previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp's) n > 1024 to the first five or six prime bases.
Abstract: Define ψm to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψm, we will have, for integers n > ψm, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψm are known for 1 ≤ m ≤ 8. Upper bounds for ψg, ψ10 and ψ11 were first given by Jaeschke, and those for ψ10 and ψ11 were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863-872).In this paper, we first follow the first author's previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp's) n > 1024 to the first five or six prime bases, which have the form n = pq with p,q odd primes and q - 1 = k(p-1), k = 4/3, 5/2, 3/2, 6; then we tabulate all Carmichael numbers > 1020, to the first six prime bases up to 13, which have the form n = q1q2q3 with each prime factor qi ≡ 3 mod 4. There are in total 36 such Carmichael numbers, 12 numbers of which are also spsp's to base 17; 5 numbers are spsp's to bases 17 and 19; one number is an spsp to the first 11 prime bases up to 31. As a result the upper bounds for ψ9, ψ10 and ψ11 are lowered from 20- and 22-decimal-digit numbers to a 19-decimal-digit number: ψ9 ≤ ψ10 ≤ ψ11 ≤ Q11 = 3825 12305 65464 13051 (19 digits) = 149491 ċ 747451 ċ 34233211. We conjecture that ψ9 = ψ10 = ψ11 = 3825 12305 65464 13051, and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor q3 and propose necessary conditions on n to be a strong pseudoprime to the first 5 prime bases. Comparisons of effectiveness with Arnault's, Bleichenbacher's, Jaeschke's, and Pinch's methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given.
TL;DR: This paper presents an alternative method to decide quickly whether a large number n is composite or probably prime, both based on the ideas of Pomerance, Baillie, Selfridge, and Wagstaff, and on a suitable combination of square, third, and fourth root testing conditions.
Abstract: . The workhorse of most compositeness tests is Miller—Rabin, which works very fast in practice, but may fail for one-quarter of all bases. We present an alternative method to decide quickly whether a large number n is composite or probably prime. Our algorithm is both based on the ideas of Pomerance, Baillie, Selfridge, and Wagstaff, and on a suitable combination of square, third, and fourth root testing conditions. A composite number n ≡ 3 mod 4 will pass our test with probability less than 1/331,000, in the worst case. For most numbers, the failure rate is even smaller. Depending on the the respective residue classes n modulo 3 and 8 , we prove a worst-case failure rate of less than 1/5,300,000, 1/480,000, and 1/331,000, respectively, for any iteration of our test. Along with some fixed precomputation, our test has running time about three times the time as for the Miller—Rabin test. Implementation can be achieved very efficiently by naive arithmetic only.
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TL;DR: A simple method for doubling the speed of safe prime generation is introduced, particularly suited to settings where a large number of RSA moduli must be generated.
Abstract: Safe primes are prime numbers of the form p = 2q +1 where q is prime. This note introduces a simple method for doubling the speed of safe prime generation. The method is particularly suited to settings where a large number of RSA moduli must be generated. keywords : safe primes, key-generation, prime-generation, RSA.