Diophantine Equations Involving Arithmetic Functions of Factorials

by Baczkowski, Daniel M.

We examine and classify the solutions to certain Diophantine equations involving factorials and some well known arithmetic functions. F. Luca has showed that there are finitely many solutions to the equation: f(n!)=a m! where f is one of the arithmetic functions ? or ? (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 ? is prime and k>0 , then ?(n!)=? k m! and ? k f(n!)=m! have finitely many solutions (?,k,m,n) , too.