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Decomposing a Factorial into Large Factors

I’ve just uploaded to the arXiv the paper “Decomposing a factorial into large factors”. This paper studies the quantity $latex {t(N)}&fg=000000$, defined as the largest quanti…

At one point, Erdös, Selfridge, and Straus claimed that this upper bound was asymptotically sharp, in the sense that as ; informally, this means we can split into factors that are (mostly) approximately the same size, when is large. However, it may be possible to now crowdsource the verification of (ii) and (iii) by supplying a suitable set of factorizations to cover medium sized , combined with some effective version of the lower bound argument that can establish for all past a certain threshold. The value singled out by Guy and Selfridge appears to be quite a suitable test case: the constructions I tried fell just a little short of the conjectured threshold of , but it seems barely within reach that a sufficiently efficient rearrangement of factors can work here.

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