14 research outputs found
The Pseudosquares Prime Sieve
We present the pseudosquares prime sieve, which finds all primes up to n
On Pseudopoints of Algebraic Curves
Following Kraitchik and Lehmer, we say that a positive integer is an -pseudosquare if it is a quadratic residue for each odd prime , yet is not a square. We extend this defintion to algebraic curves and say
that is an -pseudopoint of a curve (where )
if for all sufficiently large primes the congruence is satisfied for some .
We use the Bombieri bound of exponential sums along a curve to estimate the
smallest -pseudopoint, which shows the limitations of the modular approach
to searching for points on curves
Two Compact Incremental Prime Sieves
A prime sieve is an algorithm that finds the primes up to a bound . We say
that a prime sieve is incremental, if it can quickly determine if is
prime after having found all primes up to . We say a sieve is compact if it
uses roughly space or less. In this paper we present two new
results:
(1) We describe the rolling sieve, a practical, incremental prime sieve that
takes time and bits of space, and
(2) We show how to modify the sieve of Atkin and Bernstein (2004) to obtain a
sieve that is simultaneously sublinear, compact, and incremental.
The second result solves an open problem given by Paul Pritchard in 1994
Sieving for pseudosquares and pseudocubes in parallel using doubly-focused enumeration and wheel datastructures
We extend the known tables of pseudosquares and pseudocubes, discuss the
implications of these new data on the conjectured distribution of pseudosquares
and pseudocubes, and present the details of the algorithm used to do this work.
Our algorithm is based on the space-saving wheel data structure combined with
doubly-focused enumeration, run in parallel on a cluster supercomputer
Approximately counting semismooth integers
An integer is -semismooth if where is an integer with
all prime divisors and is 1 or a prime . arge quantities of
semismooth integers are utilized in modern integer factoring algorithms, such
as the number field sieve, that incorporate the so-called large prime variant.
Thus, it is useful for factoring practitioners to be able to estimate the value
of , the number of -semismooth integers up to , so that
they can better set algorithm parameters and minimize running times, which
could be weeks or months on a cluster supercomputer. In this paper, we explore
several algorithms to approximate using a generalization of
Buchstab's identity with numeric integration.Comment: To appear in ISSAC 2013, Boston M
The Knapsack Subproblem of the Algorithm to Compute the Erdos-Selfridge Function
This thesis summarizes the methodology of a new algorithm to compute the Erdos-Selfridge function which uses a wheel sieve, shows that a knapsack algorithm can be used to minimize the work needed to compute these values by selecting a subset of rings for use in the wheel, and compares the results of several different knapsack algorithms in this particular scenario
The I/O Complexity of Computing Prime Tables
International audienceWe revisit classical sieves for computing primes and analyze their performance in the external-memory model. Most prior sieves are analyzed in the RAM model, where the focus is on minimizing both the total number of operations and the size of the working set. The hope is that if the working set fits in RAM, then the sieve will have good I/O performance, though such an outcome is by no means guaranteed by a small working-set size. We analyze our algorithms directly in terms of I/Os and operations. In the external-memory model, permutation can be the most expensive aspect of sieving, in contrast to the RAM model, where permutations are trivial. We show how to implement classical sieves so that they have both good I/O performance and good RAM performance, even when the problem size N becomes huge—even superpolynomially larger than RAM. Towards this goal, we give two I/O-efficient priority queues that are optimized for the operations incurred by these sieves