Sieving for Pseudosquares and Pseudocubes in Parallel Using Doubly-Focused Enumeration and Wheel Datastructures

Abstract

Abstract. 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 spacesaving wheel data structure combined with doubly-focused enumeration, run in parallel on a cluster supercomputer.

Two algorithms to find primes in patterns

“…(2) Does it make sense to use Bernstein's doubly-focused enumeration to attempt to further reduce the running time? See [5,28,30] (3) A natural extension to our algorithms here is to allow the linear polynomials f i to potentially be higher degree, irreducible polynomials. See Schinzel's Hypothesis H (See [26] and [7, §1.2.2]) and the Bateman-Horn conjecture [4].…”

“…This algorithm extends the Atkin-Bernstein prime sieve with our spacesaving wheel sieve. See [27,28,29].…”

“…See [27] for a sublinear time algorithm to find all needed pseudosquares. In practice, all pseudosquares up to 10 25 are known [28]. The cost in time and space to build the wheel is, up to a constant factor, the same.…”

Strong pseudoprimes to twelve prime bases

“…Ideally, we would like to include every prime in ν for the best performance, but with normal sieving, the sieve modulus becomes quite large and requires we keep track of too many residue classes to be practical. Instead, we adapted the spacesaving wheel datastructure [16,15], which had been used successfully to sieve for pseudosquares. Sieving for primes p t with specified quadratic character modulo a list of small primes is the same algorithmic problem.…”

An Algorithm and Estimates for the Erdős-Selfridge Function (work in progress)

“…New Results. We adapted the sieving techniques from [16,12] to use the space-saving wheel sieve, which was described in [17], and was used previously to find pseudosquares [18], pseudoprimes [19], and primes in patterns [20]. Our resulting algorithm has, so far, verified all previous computations for g(k), and extended them for all k ≤ 323.…”