Decomposing a factorial into large factors

Abstract

Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N=1/e-o(1)$ was apparently established in unpublished work of Erdős, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic $$\frac{t(N)}{N}=\frac{1}{e}-\frac{c_0}{\log N}+O\left(\frac{1}{{\log }^{1+c}N}\right)$$ for an explicit constant $c_0=0.30441901\dots$ and some absolute constant $c>0$, answering a question of Erdős and Graham. For the upper bound, a further lower order term in the asymptotic expansion is also obtained. With numerical assistance, we obtain highly precise computations of $t(N)$ for wide ranges of $N$, establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that $t(N)\ge N/3$ for $N\ge 43632$, with the threshold shown to be best possible.

63 pages, 18 figures. Referee comments incorporated