The straight line complexity of small factorials and primorials

TL;DR

This paper computes the minimal straight-line programs needed to express small factorials and primorials, providing exact values for small cases and bounds for larger ones through exhaustive computer searches.

Contribution

It determines the straight-line complexity of factorials up to 22 and primorials up to 31, offering bounds for larger values using computational methods.

Findings

Exact complexities for n! with n ≤ 22

Exact complexities for primorials up to p=31

Bounds for complexities of larger factorials and primorials

Abstract

In this paper we determine the straight-line complexity of $n!$ for $n\leq 22$ and give bounds for the complexities up to $n=46$. In the same way we determine the straight-line complexity of the product of the first primes up to $p=31$ and gives bounds for $p\leq 43$. Our results are based on an exhaustive computer search of the short length straight-line programs.