Smooth Values of Quadratic Polynomials

Let Q(a) ozthorn be the set of z-smooth numbers of the form q(2) + a. It is not obvious, but this is a finite set. The cardinality can be quite large; for example, vertical bar Q(1) (1900)vertical bar >= 646890. We have a remarkably simple and fast algorithm that for any a and any z yields a subset Q(a) (z) subset of Q(a) (z) which we believe contains all but a tiny fraction of the elements of Q(a) (z), i.e. vertical bar Q(a) (z)vertical bar = (1 + o (1))vertical bar Q(a) (z)vertical bar. We have used this algorithm to compute Q(a) (500) for all 0 < a <= 25. Analyzing these sets has led to several conjectures. One is that the set of logarithms of the elements of Q(a) (z) become normally distributed for any fixed a as z -> infinity. A second has to do with the prime divisors p <= z of the sets Q(a) (z). Clearly any prime divisor p of an element of Q(a) (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 Q(a) (z) are divisible by p. Instead we conjecture that around c(p,a,z)/root p p of the elements are divisible by p where c(p,a,z) is usually between 1 and 2.