Prime number conjectures from the Shapiro class structure

Hartosh Singh Bal; Gaurav Bhatnagar

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arXiv · 1903.09619

Prime number conjectures from the Shapiro class structure

Hartosh Singh Bal·Gaurav Bhatnagar

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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.

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Hartosh Singh Bal, Gaurav Bhatnagar. 2019-03-22. Prime number conjectures from the Shapiro class structure. https://arxiv.org/abs/1903.09619

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