Conjectures about Primes and Cyclic Numbers

TL;DR

This paper explores properties of cyclic numbers, proposes conjectures analogous to those about primes, and tests these conjectures using a large dataset, leading to disproof of one such cyclic conjecture.

Contribution

It introduces new conjectures about cyclic numbers inspired by prime properties and tests them empirically using extensive computational data.

Findings

Disproved the cyclic analog of Hardy-Littlewood's second conjecture

Proposed several new conjectures about cyclic numbers

Tested conjectures on over 28 million cyclic numbers less than 10^8

Abstract

A positive integer $n$ is defined to be cyclic if and only if every group of size $n$ is cyclic. Equivalently, $n$ is cyclic if and only if $n$ is relatively prime to the number of positive integers less than $n$ that are relatively prime to $n$. Because every prime number is cyclic, it is natural to ask whether a (proved or conjectured) property of primes extends to cyclic numbers. I review proved or conjectured properties of primes (including some new conjectures about primes) and propose analogous conjectures about cyclic numbers. Using the 28,488,167 cyclic numbers less than $10^8$, I test the conjectures about cyclic numbers and disprove the cyclic analog of the second conjecture about primes of Hardy and Littlewood. Proofs or disproofs of the remaining conjectures are invited.