Remark on factorials that are products of factorials

In a paper published in 1993, Erdös proved that if n! = a! b!, where 1 < a ≤ b, then the difference between n and b does not exceed 5 log log n for large enough n. In the present paper, we improve this upper bound to ((1 + ε)/ log 2) log log n and generalize it to the equation a ₁!a ₂! ... a _k ! = n!. In a recent paper, F. Luca proved that n ? b = 1 for large enough n provided that the ABC-hypothesis holds.