Pillai showed in 1929 that the function $A(n)$ giving the number solutions of the equation $\phi(x)=n$ is unbounded in (S. Pillai, [On some functions connected with $\varphi(n)$](https://doi.org/10.1090/S0002-9904-1929-04799-2), Bull. Amer. Math. Soc. 35 (1929), 832–836). I'm interested to know about bounds on solutions of $\phi(x)=n!$ which is assigned [A055506 in OEIS](https://oeis.org/A055506), where it is claimed that if $\phi(x) = n!$, then $x$ must be a product of primes $p$ such that $p - 1$ divides $n!$. It is unclear to me if this allows me to prove that there are finitely many solutions of the equation $\phi(x) = n!$. Probably an equivalent question is to ask: is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound ?


**Related question**:
<https://math.stackexchange.com/q/3747571/156150>



Is the number of solutions of $\phi(x)=n!$ bounded? If yes, what is its bound?
**UPD.** Bound simplified.


Here is a constructive bound for the number of solutions to $\phi(x)=m$.


Let $\varphi(a) = m$. If $p^k\mid a$ for some $k\geq 1$, then $p^{k-1}(p-1)\mid m$, and thus $k\leq 1+\frac{\log(m)}{\log(p)}\leq 1+\frac{\log(m)}{\log(2)}$.
Then the number of such $a$ is bounded by
$$\prod\_{d\mid m} (2+\frac{\log(m)}{\log(2)}) = (2+\frac{\log(m)}{\log(2)})^{\tau(m)}.$$


For $m>40$, we have $2+\frac{\log(m)}{\log(2)}\leq 2\log(m)$, and thus we generously bound the number of solutions by
$$(2\log(m))^m.$$


