Timeline for "Radical" Catalan numbers?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 13, 2021 at 15:05 | vote | accept | T. Amdeberhan | ||
| Jul 11, 2021 at 15:34 | answer | added | Peter Taylor | timeline score: 3 | |
| Jul 11, 2021 at 6:46 | comment | added | Peter Taylor | @WillSawin, aha, thank you. I realised after I'd gone to bed that something was wrong because the two results as quoted were contradictory. | |
| Jul 11, 2021 at 0:00 | comment | added | Gerry Myerson | It is reported at primepuzzles.net/problems/prob_043.htm that $C_n$ is not squarefree for any $n$, $35<n\le2^{30}-1$, so it seems quite plausible that $C_{35}$ is the last squarefree Catalan number. | |
| Jul 10, 2021 at 23:53 | comment | added | Gerry Myerson | It may take a while for these estimates to kick in. benvitalenum3ers.wordpress.com/2012/03/10/catalan-num3ers factors the first $35$ Catalan numbers, and they are squarefree for $n=1,2,3,4,5,7,8,9,11,17,19,31,35$. | |
| Jul 10, 2021 at 23:39 | history | edited | Gerry Myerson | CC BY-SA 4.0 |
edited in accord with comments
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| Jul 10, 2021 at 23:10 | comment | added | Will Sawin | @PeterTaylor You seem to have dropped an exp or log in the first result. The correct statement is that the square root of the square part is $e^{ (c\pm e) \sqrt{n}$. This answers the question: As soon as this is larger than $2n+1$, as it will be for all large $n$, $C_n$ cannot be squarefree. | |
| Jul 10, 2021 at 22:57 | comment | added | Peter Taylor | In fact, Granville, A., & Ramaré, O. (1996). Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika, 43(1), 73-107 go further: when $n > 2^{1617}$, $\binom{2n}{n}$ is divisible by the square of some prime $p > \sqrt{n}$. | |
| Jul 10, 2021 at 22:49 | comment | added | Wojowu | What does "square-free primes" mean? | |
| Jul 10, 2021 at 22:47 | comment | added | Peter Taylor | Seems unlikely based on the main theorem of Sárközy, A. (1985). On divisors of binomial coefficients, I. Journal of Number Theory, 20(1), 70-80: the square root of the largest square number dividing $\binom{2n}{n}$ is in the range $(c \pm \varepsilon)\sqrt n$ for all $n$ greater than a lower bound which depends on $\varepsilon$, where $c \approx 0.855$. | |
| Jul 10, 2021 at 22:47 | comment | added | JoshuaZ | I think you mean just "square-free" rather than "square free primes." | |
| Jul 10, 2021 at 21:29 | history | asked | T. Amdeberhan | CC BY-SA 4.0 |