On the ultimate complexity of factorials - Cheng, Qi (Theoretical computer science, 2004)
It has long been observed that certain factorization algorithms provide a way to write the product of many different integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (i.e. n ! ) by straight-line programs. Formally, we say that a sequence of integers a n is ultimately f ( n ) -computable, if there exists a nonzero integer sequence m n such that for any n , a n m n can be computed by a straight-line program (using only additions, subtractions and multiplications) of length at most f ( n ) . Shub and Smale [12] showed that if n ! is ultimately hard to compute, then the algebraic version of NP ≠ P is true. Assuming a widely believed number theory conjecture concerning smooth numbers in a short interval, a subexponential upper bound ( exp ( c log n log log n ) ) for the ultimate complexity of n ! is proved in this paper, and a randomized subexponential algorithm constructing such a short straight-line program is presented as well.