Terence Tao @tao@mathstodon.xyz
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It is intuitively plausible that formalization can help detect errors in mathematical papers. I discovered today that more old-fashioned numerics can also achieve a similar goal. In a paper I just uploaded at arxiv.org/abs/2503.20170 , I claimed an upper bound on a certain number-theoretic function t(N), and gave a proof. But, inspired by a nascent crowdsourced effort at terrytao.wordpress.com/2025/03 to get enough estimates on this quantity for medium values of N to verify some explicit conjectures of Guy and Selfridge, I decided to plot this upper bound against the known values of t(N), and found a few places where the upper bound was actually *smaller* than the known value. This was of course distressing, but by isolating the smallest counterexample and numerically verifying key intermediate claims in my proof, I found where the error was (I was "off by one" in a certain estimate involving the floor function). This will be patched in the next revision of the paper of course, but is another example of how computer assistance (this time in the form of traditional numerics) can help detect and fix errors in papers.

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Decomposing a factorial into large factors

Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ numbers 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 The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erdős, Selfridge, and Strauss; but the proof is lost. Here we obtain the more precise asymptotic bounds $$ \frac{1}{e} - \frac{O(1)}{\log N} \leq \frac{t(N)}{N} \leq \frac{1}{e} - \frac{c_0+o(1)}{\log N}$$ for some explicit constant $c_0 \approx 0.3044$, answering a question of Erdős and Graham. Our methods are elementary, aside from the use of the prime number theorem (with classical error term). Using an effective version of the upper bound argument, we also show that $t(N)/N < 1/e$ for $N \neq 1,2,4$, answering a question of Guy and Selfridge.

arxiv.org · arXiv.org

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