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Decomposing a factorial into large factors

Boris Alexeev, Evan Conway, Matthieu Rosenfeld, Andrew V. Sutherland, Terence Tao, Markus Uhr, Kevin Ventullo

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 The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erdős, Selfridge, and Straus; but the proof is lost.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 $t(N) \geq N/3$ for $N \geq 43632$, with the threshold shown to be best possible.

Decomposing a factorial into large factors

Abstract

Let denote the largest number such that can be expressed as the product of integers greater than or equal to . The bound was apparently established in unpublished work of Erdős, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic for an explicit constant and some absolute constant , 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 for wide ranges of , establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that for , with the threshold shown to be best possible.

Paper Structure

This paper contains 23 sections, 23 theorems, 322 equations, 18 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Linear and integer programming solvers
  3. Author contributions and data
  4. Acknowledgments
  5. Notation and basic estimates
  6. Approximation by 3-smooth numbers
  7. Sums over primes
  8. Greedy algorithms
  9. Implementation of the greedy algorithm
  10. A fast variant of the greedy algorithm
  11. Optimizing bounds produced by the greedy algorithm
  12. Proving Theorem 1.3(iii) for 67425 <= N <= 1e14
  13. Linear programming
  14. Some upper bounds
  15. Rearranging the standard factorization
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $N$ be a natural number. $\blacktriangleleft$$\blacktriangleleft$

Figures (18)

  • Figure 1: The function $t(N)/N$ (blue) for $N \leq 200$, using the data from https://oeis.org/A034258, as well as the trivial upper bound $(N!)^{1/N}/N$ (green), the improved upper bound from \ref{['upper-crit']} (pink), which is asymptotic to \ref{['asym']} (purple), and the function $\lfloor 2N/7 \rfloor/N$ (brown), which we show to be a lower bound for $N \neq 56$. \ref{['main']}(iv) implies that $t(N)/N$ is asymptotic to \ref{['asym']} (purple), which in turn converges to $1/e$ (orange), although we believe that \ref{['tna']} (gray) asymptotically becomes a sharper approximation. The threshold $1/3$ (red) is permanently crossed at $N=43632$.
  • Figure 2: A continuation of \ref{['fig1']} to the region $80 \leq N \leq 599$.
  • Figure 3: The piecewise continuous function $x\mapsto \frac{1}{e} f_e(x)$, together with its mean value $c_0 = 0.30441901\dots$ and the upper bound $\log(1+ex)/ex$. The function exhibits an oscillatory singularity at $x=0$ similar to $\sin 1/x$ (but it is always nonnegative and bounded). Informally, the function $f_e$ quantifies the difficulty that large primes in the factorization of $N!$ have in becoming only slightly larger than $N/e$ after multiplying by a natural number.
  • Figure 4: The function $\log \frac{\lceil x \rceil^{\langle 2,3 \rangle}}{x}$, compared against $\kappa_x$.
  • Figure 5: The function $\sum_{k \leq x}^* 1 - \frac{x}{3}$.
  • ...and 13 more figures

Theorems & Definitions (53)

  • Example 1.1
  • Remark 1.2
  • Theorem 1.3: Main theorem
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1: Approximation by $3$-smooth numbers
  • Lemma 2.2
  • proof
  • Lemma 2.3: Effective bounds for oscillatory sums over primes
  • Remark 3.1
  • ...and 43 more