F. Luca has shown that there are finitely many solutions to the equation:
f(n!)=a m!
where f is one of the arithmetic functions φ or σ (sum of the divisors function) and a is a rational number.
We study the solutions for this equation when a is a prime power or a reciprocal of a prime power.
Furthermore, we prove that if ρ is prime and k>0, then φ(n!)=ρ^k m!   and   ρ^k f(n!)=m!
have finitely many solutions (ρ,k,m,n), too.
If interested in reading the paper, then Click here!


* Note we are currently in the process of producing a more general form of the theorem above.
Namely Michael Filaseta, Ognian Trifonov and I are showing that
b f(n!)=a m!
has finitely many solutions in positive integers a, b, m, and n whenever f is one of the
arithmetic functions φ, σ_α (sum of the divisors to the power α function
note: α=0 is the divisor function and α=1 is the sum of divisors function),
or τ (Ramanujan's tau function).