Decomposing a factorial into large factors
Decomposing a factorial into large factorsLet \(t(N)\) 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 The bound \(t(N)/N = 1/e-o(1)\) was apparently established in unpublished work of Erdős, Selfridge, and Straus; but the proof is lost.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 \(t(N) \geq N/3\) for \(N \geq 43632\), with the threshold shown to be best possible.
Boris Alexeev, Evan Conway, Matthieu Rosenfeld, Andrew V. Sutherland, Terence Tao, Markus Uhr, and Kevin Ventullo
Decomposing a factorial into large factors
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