Theodore Hui, Pure Mathematics Department, University of Waterloo
“The Dynamics of the w Function”
Define P(x) the largest prime factor of x and let w(pqr)=P(p+q)P(p+r)P(q+r) for primes p,q,r. Then w maps the set A3 = pqr: not all equal to itself. It is also known that under the iteration of w, every element in A3 would eventually goes to the cycle 20,98,63,75. For any n ∈ A3, we use ind(n) to denote the least nonnegative integer i such that wi(n) ∈ 20,98,63,75. We also use π(x) to denote the number of primes less than or equal to x. Then one can show that ind(n)=O(log π(P(n))). Moreover, one can employ The Green-Tao’s Theorem on Arithmetic Progressions in the Primes to show that ind(n) can be arbitrarily large. If time allows, several generalizations of the w function will also be mentioned.