A note on Deaconescu’s conjecture

Abstract

Hasanalizade [5] studied Deaconescu’s conjecture for positive composite integer n. A positive composite integer n ≥ 4 is said to be a Deaconescu number if S2(n) | ϕ(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., ω(n) ≥ 17 and must be strictly larger than 5.86 · 1022. Further, we prove that if any Deaconescu number n has all prime divisors greater than or equal to 11, then ω(n) ≥ p*, where p* is the smallest prime divisor of n and if n ∈ D3 then all the prime divisors of n must be congruent to 2 modulo 3 and ω(n) ≥ 48.

Keywords

• euler’s totient function
• schemmel’s totient function
• deaconescu’s conjecture
• 11a25