Draft:Deaconescu's schemmel's totient problem: Difference between revisions

In mathematics, schemmel's totient function $S_{2}(n)$ is a multiplicative function defined by

$S_{2}(p^{\alpha })={\begin{cases}0,&{\text{if }}p=2\\p^{\alpha -1}(p-2),&{\text{if }}p>2\end{cases}}$

for all primes $p$ and positive integers $\alpha$. In 2000 Deaconescu conjectured that for $n\geq 2$

$S_{2}(n)\mid \varphi (n)-1$

if and only if $n$ is a prime. Clearly, the conjecture states that for every $M\geq 1$, the set $D_{M}$ of integers satisfying

$MS_{2}(n)=\varphi (n)-1$

contains only prime number. It is easy to verify that for any prime $p$ we have $S_{2}(p)\mid \varphi (p)-1$. A composite integer $n$ is a Deaconescu number (or has the Deaconescu property) if it satisfies $MS_{2}(n)=\varphi (n)-1$. Hasanalizade proved that a Deaconescu number $n$ must be an odd, squarefree positive integer such that $\omega (n)\geq 7$ and

$n<2^{2^{\omega (n)}+\omega (n)}-2^{2^{\omega (n)-1}+\omega (n)}$.

Further, he proved an analogous result due to Hernandez and Luca that for a monic non-constant polynomial $P(X)\in \mathbb {Z} [X]$, there are at most finitely many composite integers $n$ such that $S_{2}(n)\mid \varphi (n)-1$ and $P(S_{2}(n))\equiv 0{\pmod {\phi (n)}}$. Mandal improved Hasanalizade's result by proving that a Deaconescu number $n$ must have $\omega (n)\geq 17$ and must be strictly larger than $5.86\times 10^{22}$. Further, he proved 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 $\omega (n)\geq 48$.