On the ultimate complexity of factorials

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 sequenc...

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Bibliographic Details
Published in:Theoretical computer science Vol. 326; no. 1; pp. 419 - 429
Main Author: Cheng, Qi
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 20.10.2004
Elsevier
Subjects:
Algebra
Algorithmics. Computability. Computer arithmetics
Applied sciences
Computational and structural complexity
Computer science; control theory; systems
Exact sciences and technology
Mathematics
Number theory
Sciences and techniques of general use
Theoretical computing
Computational and structural complexity
Integer
Upper bound
Computer theory
Structural complexity
Computational complexity
Factorization
Number theory
ISSN:0304-3975, 1879-2294
Online Access:Get full text
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Summary: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.
ISSN:0304-3975
1879-2294
DOI:10.1016/j.tcs.2004.06.020