Two combinatorial applications of smooth numbers

Series
Combinatorics Seminar
Time
Tuesday, February 10, 2015 - 12:00pm for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nathan McNew – Dartmouth College
Organizer
Ernie Croot
We look at two combinatorial problems which can be solvedusing careful estimates for the distribution of smooth numbers. Thefirst is the Ramsey-theoretic problem to determine the maximal size ofa subset of of integers containing no 3-term geometric progressions.This problem was first considered by Rankin, who constructed such asubset with density about 0.719. By considering progressions among thesmooth numbers, we demonstrate a method to effectively compute thegreatest possible upper density of a geometric-progression-free set.Second, we consider the problem of determining which prime numberoccurs most frequently as the largest prime divisor on the interval[2,x], as well as the set prime numbers which ever have this propertyfor some value of x, a problem closely related to the analysis offactoring algorithms.