A Note on Deaconescu's Conjecture

TL;DR

This paper advances the understanding of Deaconescu's conjecture by establishing lower bounds on the number of prime divisors and size of Deaconescu numbers, with specific conditions on their prime factors.

Contribution

It improves previous results by proving that Deaconescu numbers must have at least 17 prime divisors and be larger than 5.86×10^{22}, also exploring prime divisor conditions.

Findings

Deaconescu numbers have at least 17 prime divisors.

Such numbers are larger than 5.86×10^{22}.

If all prime divisors ≥ 11, then the number of prime divisors is at least the smallest prime divisor.

Abstract

Hasanalizade [1] studied Deaconescu's conjecture for positive composite integer $n$. A positive composite integer $n\geq4$ is said to be a Deaconescu number if $S_2(n)\mid \phi(n)-1$. In this paper, we improve Hasanalizade's result by proving that a Deaconescu number $n$ must have at least seventeen distinct prime divisors, i.e., $\omega(n)\geq 17$ and must be strictly larger than $5.86\cdot10^{22}$. Further, we prove that if any Deaconescu number $n$ has all prime divisors greater than or equal to $11$, then $\omega(n)\geq p^{*}$, where $p^{*}$ is the smallest prime divisor of $n$ and if $n\in D_3$ then all the prime divisors of $n$ must be congruent to $2$ modulo $3$ and $\omega(n)\geq 48$.