On the equation n!=a1!a2!⋯at!n!=a1!a2!⋯at!

For a given positive integer nn, we consider positive integers a1,a2…,ata1,a2…,at such that a1!a2!⋯at!=n!a1!a2!⋯at!=n!. Luca proved that n−a1=1n−a1=1 if abcabc conjecture holds and nn is sufficiently large. Erdős, Bhat and Ramachandra gave unconditional upper bounds on n−a1n−a1 which we improve in this paper. Further we solve the equation when P(n+1)≤79P(n+1)≤79 by using our estimate.