Ingredient one: In the MathOverflow Question
How Do These Primes Jump?
I consider a basic algorithm S which is a
resource-constrained Sieve of Eratosthenes. Follow the link
for the details; the idea is to process the positive numbers
greater than 1 in increasing order so that for every prime
(uncovered number), one acquires a new stone labelled with that prime,
and places the stone (covers) on the next uncovered multiple of that prime (which
can be proved to be twice that prime), and so that for every
composite (a covered number) one picks up the stone that covers the composite and has label p
on it, and move it the least multiple of p distance greater than the composite
to a number that is not covered. The question considers trajectories of stones given
the constraints that no number ever gets covered by more than one stone, and
that the numbers are processed serially in increasing order.
One result of this is a mapping from natural numbers n greater than 1 to primes
$q_n$ such that $q_n$ divides n. There is an answer which is a partial
analysis of the behaviour of the trajectories.
