Prime number conjectures from the Shapiro class structure

When/Where:

January 22, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

The height \(H(n)\) of \(n\), introduced by Pillai in 1929, is the smallest positive integer \(i\) such that the \(i\)-th iterate of Euler’s totient function at \(n\) is 1. H. N.Shapiro (1943) studied the structure of the set of all numbers at a particular height. We provide a formula for the height function thereby extending a result of Shapiro. We list steps to generate numbers of any height which turns out to be a useful way to think about this construct. We present some theoretical and computational evidence to show that \(H\) and its relatives are closely related to the important functions of number theory, namely \(\pi(n)\) and the \(n\)-th prime \(p_n\). This is joint work with Hartosh Singh Bal.