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Original Articles

Smooth Values of Quadratic Polynomials

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Pages 447-452 | Published online: 02 Apr 2019
 

Abstract

Let Qa(z) be the set of z-smooth numbers of the form q2+a. It is not obvious, but this is a finite set. The cardinality can be quite large; for example, |Q1(1900)|646890. We have a remarkably simple and fast algorithm that for any a and any z yields a subset Qa(z)Qa(z) which we believe contains all but a tiny fraction of the elements of Qa(z), i.e. |Qa(z)|=(1+o(1))|Qa(z)|. We have used this algorithm to compute Qa(500) for all 0<a25. Analyzing these sets has led to several conjectures. One is that the set of logarithms of the elements of Qa(z) become normally distributed for any fixed a as z. A second has to do with the prime divisors pz of the sets Qa(z). Clearly any prime divisor p of an element of Qa(z) must have the property that – a is a square modulo p. For such a p we might naively expect that approximately 2/p of the elements of Qa(z) are divisible by p. Instead we conjecture that around cp,a,z/p of the elements are divisible by p where cp,a,z is usually between 1 and 2.

Additional information

Funding

The first author was supported in part by a grant from the NSF.

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