# Prime number conjectures from the Shapiro class structure

## TL;DR

This paper explores the height function related to Euler's totient iterates, extending Shapiro's work, and presents conjectures linking it to prime counting functions and prime numbers.

## Contribution

It provides a formula for the height function, extends results on odd heights, and conjectures new relationships between height and prime number functions.

## Key findings

- Extended results on the largest odd numbers at a height
- Formulated conjectures relating height to prime counting functions
- Presented computational evidence supporting these conjectures

## 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 height. We state a formula for the height function due to Shapiro and use it to list steps to generate numbers at any height. This turns out to be a useful way to think of this construct. In particular, we extend some results of Shapiro regarding the largest odd numbers at a height. 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 $p_n$. We conjecture formulas for $\pi(n)$ and $p_n$ in terms of the height function.