Doubly focused enumeration of locally square polynomial values

Abstract

Abstract. Let f be a nonconstant squarefree 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 → ∞, 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 record-setting 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.

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

“…This, then, motivates the search for larger and larger peudosquares and pseudocubes, and attempts to predict their distribution. See, for example, Wooding and Williams [12] and also [7,11,8,2,10].…”

“…We discuss the implications of this new data on the conjectured distribution of pseudosquares and pseudocubes in §3, and give a minor refinement of the current conjectures. Then we describe our parallel algorithm, based on Bernstein's doubly-focused enumeration [2], which is used in a way similar, but not identical to the work of Wooding and Williams [12], combined with the space-saving wheel data structure presented in [9, §4.1]. We then suggest ideas for future work in §5.…”

On pseudopoints of algebraic curves

“…Furthermore for a real x ≥ 1, we denote by P f (x) = P f ∩ [2, x] and say that an integer n ≥ 0 is an x-pseudopoint of f if for all p ∈ P f (x) we have (n, m) ∈ Z f (p) for some m, but the equation f (n, m) = 0 has no integer solution m ∈ Z. We note that Bernstein [2] has introduced and studied this notion in the case of the polynomials of the form…”