Results 1  10
of
12
Some results on pseudosquares
 Math. Comp
, 1996
"... Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
Abstract. If p is an odd prime, the pseudosquare Lp is defined to be the least positive nonsquare integer such that Lp ≡ 1 (mod 8) and the Legendre symbol (Lp/q) = 1 for all odd primes q ≤ p. In this paper we first discuss the connection between pseudosquares and primality testing. We then describe a new numerical sieving device which was used to extend the table of known pseudosquares up to L271. We also present several numerical results concerning the growth rate of the pseudosquares, results which so far confirm that Lp √ e p/2, an inequality that must hold under the extended Riemann Hypothesis. 1.
An Investigation of Bounds for the Regulator of Quadratic Fields
 Experimental Mathematics
, 1995
"... This paper considers the following problems: How large, and how small, can R get? And how often? The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large ..."
Abstract

Cited by 12 (6 self)
 Add to MetaCart
This paper considers the following problems: How large, and how small, can R get? And how often? The answer is simple for the problem of how small R can be, but seems to be extremely difficult for the question of how large
A Problem Concerning a Character Sum
, 1999
"... this paper we exhibit some techniques which were successful in producing, for each k such that 3 k 2000, a value for p such that S(k) > 0. 1. INTRODUCTION ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
this paper we exhibit some techniques which were successful in producing, for each k such that 3 k 2000, a value for p such that S(k) > 0. 1. INTRODUCTION
Experimental results on class groups of real quadratic fields (Extended Abstract)
 ALGORITHMIC NUMBER THEORY  ANTSIII, LECTURE NOTES IN COMPUTER SCIENCE 1423
, 1998
"... In an effort to expand the body of numerical data for real quadratic fields, we have computed the class groups and regulators of all real quadratic fields with discriminant ∆<10 9. We implemented a variation of the group structure algorithm for general finite Abelian groups described in [2] in the C ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
In an effort to expand the body of numerical data for real quadratic fields, we have computed the class groups and regulators of all real quadratic fields with discriminant ∆<10 9. We implemented a variation of the group structure algorithm for general finite Abelian groups described in [2] in the C++ programming language using builtin types together with a few routines from the LiDIA system [12]. This algorithm will be described in more detail in a forthcoming paper. The class groups and regulators of all 303963581 real quadratic fields were computed on 20 workstations (SPARCclassics, SPARC4’s, and SPARCultra’s) by executing the computation for discriminants in intervals of length 10 5 on single machines and distributing the overall computation using PVM [8]. The entire computation took just under 246 days of CPU time (approximately 3 months real time), an average of 0.07 seconds per field. In this contribution, we present the results of this experiment, including data supporting the truth of Littlewood’s bounds on the function L (1,χ∆) [13]and Bach’s bound on the maximum norm of the prime ideals required to generate the class group [1]. Data supporting several of the CohenLenstra heuristics [6,7] is presented, including results on the percentage of noncyclic odd parts of class groups, percentages of odd parts of class numbers equal to small odd integers, and percentages of class numbers divisible by small primes p. We also give new examples of irregular class groups, including examples for primes p ≤ 23 and one example of a rank 3 5Sylow subgroup (3 noncyclic factors), the first example of a real quadratic class group which has a pSylow subgroup with rank greater than 2 and p>3. 1 The L (1,χ¡) Function Much interest has been shown in extreme values of the L (1,χ∆) function [3,14,10,4]. A result of Littlewood [13] and Shanks [14] shows that under the
Results and estimates on pseudopowers
 Math. Comp
, 1996
"... Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a power ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a powerof2. Next, we define an xpseudopower of the base 2tobeapositiveintegern that is not a power of 2, but looks like a power of 2 modulo all primes p ≤ x. Let P2(x) denote the least such n. We give an unconditional upper bound on P2(x), a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that P2(x)isaboutexp(c2x/log x) for a certain constant c2. We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than 2 are also given. 1.
NEW QUADRATIC POLYNOMIALS WITH HIGH DENSITIES OF PRIME VALUES
"... Abstract. Hardy and Littlewood’s Conjecture F implies that the asymptotic density of prime values of the polynomials fA(x) =x 2 + x + A, A ∈ Z, is related to the discriminant ∆ = 1 − 4A of fA(x) viaaquantityC(∆). The larger C(∆) is, the higher the asymptotic density of prime values for any quadrati ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. Hardy and Littlewood’s Conjecture F implies that the asymptotic density of prime values of the polynomials fA(x) =x 2 + x + A, A ∈ Z, is related to the discriminant ∆ = 1 − 4A of fA(x) viaaquantityC(∆). The larger C(∆) is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant ∆. A technique of Bach allows one to estimate C(∆) accurately for any ∆ < 0, given the class number of the imaginary quadratic order with discriminant ∆, and for any ∆> 0given the class number and regulator of the real quadratic order with discriminant ∆. The Manitoba Scalable Sieve Unit (MSSU) has shown us how to rapidly generate many discriminants ∆ for which C(∆) is potentially large, and new methods for evaluating class numbers and regulators of quadratic orders allow us to compute accurate estimates of C(∆) efficiently, even for values of ∆ 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(∆) is larger than any previously known examples. 1.
Doublyfocused enumeration of pseudosquares and pseudocubes
 In Proceedings of the 7th International Algorithmic Number Theory Symposium (ANTS VII
, 2006
"... Abstract. This paper offers numerical evidence for a conjecture that primality proving may be done in (log N) 3+o(1) operations by examining the growth rate of quantities known as pseudosquares and pseudocubes. In the process, a novel method of solving simultaneous congruences— doublyfocused enumer ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. This paper offers numerical evidence for a conjecture that primality proving may be done in (log N) 3+o(1) operations by examining the growth rate of quantities known as pseudosquares and pseudocubes. In the process, a novel method of solving simultaneous congruences— doublyfocused enumeration — is examined. This technique, first described by D. J. Bernstein, allowed us to obtain recordsetting sieve computations in software on general purpose computers. 1
Doubly Focused Enumeration Of Locally Square Polynomial Values
"... Let f be an irreducible polynomial. Which of the values f(c + 1); f(c + 2); : : : ; f(c + H) are locally square at all small primes? This paper presents an algorithm that answers this question in time H=M 2+o(1) for an average small c as H ! 1, where M = H 1=log 2 log H . In contrast, the usu ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Let f be an irreducible polynomial. Which of the values f(c + 1); f(c + 2); : : : ; f(c + H) are locally square at all small primes? This paper presents an algorithm that answers this question in time H=M 2+o(1) for an average small c as H ! 1, where M = H 1=log 2 log H . In contrast, the usual method takes time H=M 1+o(1) . This paper also presents the results of two computations: an enumeration of locally square integers up to 24 2 64 , and an enumeration of locally square values of x 3 + y 7 for small x and y.
A Problem Concerning a Character Sum (Extended Abstract)
"... ? ) E. Teske 1 and H.C. Williams ??2 1 Technische Universitat Darmstadt Institut fur Theoretische Informatik Alexanderstrae 10, 64283 Darmstadt Germany 2 University of Manitoba Dept. of Computer Science Winnipeg, MB Canada R3T 2N2 Abstract. Let p be a prime congruent to 1 modulo 4, n p ..."
Abstract
 Add to MetaCart
? ) E. Teske 1 and H.C. Williams ??2 1 Technische Universitat Darmstadt Institut fur Theoretische Informatik Alexanderstrae 10, 64283 Darmstadt Germany 2 University of Manitoba Dept. of Computer Science Winnipeg, MB Canada R3T 2N2 Abstract. Let p be a prime congruent to 1 modulo 4, n p the Legendre symbol and S(k) = P p 1 n=1 n k n p . The problem of nding a prime p such that S(3) > 0 was one of the motivating forces behind the development of several of Shanks' ideas for computing in algebraic number elds, although neither he nor D. H. and Emma Lehmer were ever successful in nding such a p. In this extended abstract we summarize some techniques which were successful in producing, for each k such that 3 k 2000, a value for p such that S(k) > 0. 1 Introduction Let d denote a fundamental discriminant of an imaginary quadratic eld IK = Q( p d ) and let h(d) denote the class number of IK. Let p be a prime ( 3(mod 4)), n p the Legendre symbol and S...