Timeline for Is $\sum_{k=1}^{n}\frac{(n-1)!}{(k-1)!}$ composite for $n\geq 4$?
Current License: CC BY-SA 4.0
9 events
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| Oct 16, 2023 at 5:04 | comment | added | Terry Tao | This is broadly consistent with the (very crude) heuristic that each $a_n$ has a "probability" of $\approx \frac{1}{\log a_n} \approx \frac{1}{n \log n}$ of being prime; the sum $\sum_{n=2}^\infty \frac{1}{n \log n}$ diverges, but only double logarithmically, so one would tentatively expect an infinite number of counterexamples, but spread out extremely thinly, and it makes sense that the first one is only seen at $n \approx 2000$. | |
| Oct 15, 2023 at 2:15 | comment | added | ho boon suan | Yes, I believe Mathematica's FactorInt function with default settings is using the Lenstra elliptic-curve factorization method when run with input $a_{2017}$, which only shows that $a_{2017}$ has no small prime factors. To check if $a_{2017}$ is prime with certainty using Mathematica, one could try using the PrimalityProving package, though it would probably take a very, very, very long time to run. | |
| Oct 14, 2023 at 22:52 | comment | added | Dave Benson | Well, Magma didn't have such an easy time of it. It very quickly decided that $a_{2000}$ to $a_{2016}$ were composite, but at $a_{2017}$ my (quite powerful) laptop sat there for five minutes getting hotter and blowing out air before I aborted the computation. I suspect that Mathematica is lying to you about its confidence in its answer. | |
| Jun 22, 2021 at 0:15 | comment | added | 2734364041 | Mathematica seems to think that $a_{2017}$ is prime. I used the FactorInteger command, and after about 10 seconds, it says that $a_{2017}$ is the only prime factor. Mathematica's factorization algorithm is usually not this efficient, which suggests that either (a) Mathematica failed fantastically, of (b) $a_{2017}$ is a prime of a special form. Maybe the algorithm works faster for numbers passing the pseudoprime test that you used. | |
| Jul 28, 2020 at 6:52 | vote | accept | solpa | ||
| Jul 27, 2020 at 18:38 | vote | accept | solpa | ||
| Jul 27, 2020 at 18:39 | |||||
| Jul 27, 2020 at 14:19 | comment | added | Robert Israel | A pity this wasn't discovered $3$ years ago. | |
| Jul 27, 2020 at 7:28 | vote | accept | solpa | ||
| Jul 27, 2020 at 7:28 | |||||
| Jul 27, 2020 at 7:13 | history | answered | Fredrik Johansson | CC BY-SA 4.0 |