Decomposing a factorial into large factors (second version)

Boris Alexeev, Evan Conway, Matthieu Rosenfeld, Andrew Sutherland, Markus Uhr, Kevin Ventullo, and I have uploaded to the arXiv a second version of our paper “Decomposing a factorial into large factors“. This is a completely rewritten and expanded version of a previous paper of the same name. Thanks to many additional theoretical and numerical contributors from the other coauthors, we now have much more precise control on the main quantity studied in this paper, allowing us to settle all the previous conjectures about this quantity in the literature.

As discussed in the previous post, denotes the largest integer such that the factorial can be expressed as a product of factors, each of which is at least . Computing is a special case of the bin covering problem, which is known to be NP-hard in general; and prior to our work, was only computed for ; we have been able to compute for all . In fact, we can get surprisingly sharp upper and lower bounds on for much larger , with a precise asymptotic

for an explicit constant , which we conjecture to be improvable to

for an explicit constant : … For instance, we can demonstrate numerically that

As a consequence of this precision, we can verify several conjectures of Guy and Selfridge, namely

Guy and Selfridge also claimed that one can establish for all large purely by rearranging factors of and from the standard factorization of , but surprisingly we found that this claim (barely) fails for all :

The accuracy of our bounds comes from several techniques:

To me, the biggest surprise was just how stunningly accurate the linear programming methods were; the very large number of repeated prime factors here actually make this discrete problem behave rather like a continuous one.