Timeline for On an attempt to create interesting variants of Firoozbakht's conjecture, evoking combinations of different generalized means
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 19, 2019 at 22:46 | vote | accept | user142929 | ||
| Dec 19, 2019 at 22:36 | answer | added | Joel Moreira | timeline score: 1 | |
| Dec 19, 2019 at 19:34 | comment | added | user142929 | Many thanks for your feedback, the differences $LHS-RHS$ are very small even for the humble segment of my experiments $2700\leq n\leq 50000$, thus I suspect that maybe some user can to find a counterexample, but the best way to refute the conjecture is provide a convincing reasoning as yours. Thus I am going to wait if there is some answer, feel free to add yourself reasoning as an answer. Many thanks @JoelMoreira | |
| Dec 19, 2019 at 16:46 | comment | added | Joel Moreira | Hm, apparently $p_n/(n\log n)$ decays to $1$ rather slowly (it behaves like $1+\frac{\log\log n}{\log n}$), so I take back my guess that an improvement of PNT will show that (2) is false. | |
| Dec 19, 2019 at 16:15 | comment | added | Joel Moreira | Replacing $p_n$ with $n\log n$ in (2) yields a false inequality for all large $n$. I'm guessing that some improved form of the prime number theorem may imply that (2) is false. | |
| Dec 19, 2019 at 13:21 | comment | added | user142929 | Due the form of the functions $\lambda(n)$ and $\mu(n)$ that I've choosen by experimenting, the inequality $(2)$ can be written as $$e^{e^{A(n)}}>n^{n^{G(n)}},\text{ for integers } n\geq 2700$$ where the exponent $A(n)$ denotes the arithmetic mean $A(\underbrace{1,\ldots,1}_{n\text{ ones}},p_{n+1})=\frac{n+p_{n+1}}{n+1}$ and $G(n)$ denotes the geometric mean $G(\underbrace{1,\ldots,1}_{n-1\text{ ones}},p_{n})=\sqrt[n]{p_{n}}$. This is just to emphasize that my inequality is a comparison of means, in the same way that Firoozbakht's conjecture can be interpreted in terms of geometric means | |
| Dec 18, 2019 at 23:40 | comment | added | user142929 | Also is welcome comments about if my inequality $(2)$ is interesting or if there are similar inequalities in the literature. | |
| Dec 18, 2019 at 21:19 | comment | added | user142929 | I think that the idea is thus if it is possible to prove or refute the inequality $(2)$ for some $n_0>2700$ using the explicit inequalities for primes, since I cann't find a counterexample. Feel free to add comments/critics about this question, I'm waiting your feedback. Thank you very much. | |
| Dec 18, 2019 at 10:44 | history | edited | user142929 | CC BY-SA 4.0 |
Fixed a typo in the code.
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| Dec 18, 2019 at 10:26 | history | edited | user142929 | CC BY-SA 4.0 |
Grammar
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| Dec 18, 2019 at 10:17 | history | asked | user142929 | CC BY-SA 4.0 |