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