A Note on Deaconescu's Conjecture

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 $\omega(n)\geq 48$.