Timeline for A Collatz-like problem on prime numbers
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
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| Oct 22, 2017 at 8:29 | comment | added | Gerry Myerson | The sequence for $p^2+1$ was posted to OEIS in 2006, oeis.org/A031439 | |
| Oct 21, 2017 at 13:10 | answer | added | Shawn Schafer | timeline score: 4 | |
| Jul 18, 2017 at 18:25 | comment | added | Sebastien Palcoux | The last four digits of $\alpha=\lambda^{-1}$ (written in a previous comment) are wrong, it is not $6997$ but $5533$. | |
| Jul 15, 2017 at 15:07 | comment | added | Michael Lugo | @YaakovBaruch: it's interesting to note that if we take $f(p)$ to be the largest prime factor of $p^2 + 1$ then iteration appears to lead to sequences that head off to infinity, as you'd expect since the Golomb-Dickman constant is greater than $1/2$. | |
| Jul 15, 2017 at 14:38 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
Computation of a prime $p$ such that $\ell(p)=r$ for a given $r$.
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| Jul 14, 2017 at 14:32 | comment | added | Sebastien Palcoux | Obviously, we get infinitely many cycles with the function $f(p)=p^2$. | |
| Jul 14, 2017 at 12:48 | comment | added | Yaakov Baruch | @SebastienPalcoux. In Collatz's problem the components of the $\mathbb N \rightarrow \mathbb N$ iterative step are simple, closely related, with clear pressure in one direction (down) and yet the tantalizing early major anomaly of 27. If not done before, it could be an excellent question for MathOverflow to ask for examples of similarly intriguing functions $\mathbb N \rightarrow \mathbb N$ - but personally I think that the one you are proposing is too contrived. Cheers | |
| Jul 14, 2017 at 12:39 | comment | added | Sebastien Palcoux | @YaakovBaruch: so a nice function to study iterations would be $f(p)= \text{gpf}(\left \lfloor{p^{\alpha}}\right \rfloor)$ or $\text{gpf}(\left \lceil{p^{\alpha}}\right \rceil)$ with $\alpha = \lambda^{-1} \simeq 1.6017170700590876997$ the inverse of the Golomb–Dickman constant. Note that $(10^{r})^{1+10^{-n}} \sim 10^{r} + r \ln(10) \cdot 10^{-n+r}$ for $r \ll 10^n$, so the above approximation of $\alpha$ would be ok if we deal with primes $p<10^{10}$. | |
| Jul 14, 2017 at 8:48 | comment | added | Yaakov Baruch | It would be surprising to find that intermediate case. The step down in the iteration is on average like raising to a 0.624... power (see Golomb-Dickman constant) which overpowers affine transformations, and is overpowered by polynomial ones of degree is $\ge 2$. Unlike in the Collatz problem, the steps up and down are drastic and of different nature (polynomial vs. arithmetic) so once both occur I'd expect things to "mix up" and follow long term expectations. A very very faint hope (in degree $\ge 2$) would be if after 1 or 2 steps up from a small prime, one hit a product of small primes... | |
| Jul 14, 2017 at 3:52 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
champion numbers
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| Jul 11, 2017 at 17:05 | comment | added | Gerhard Paseman | While not quite the same thing, you could construct some small cycles and then fit a polynomial map to them. For infinitely many cycles, consider floor or ceiling of p to a power ($e/2$ perhaps?) matching the down scaling of gpf. Gerhard "Make Up Your Own Answer" Paseman, 2017.07.11. | |
| Jul 11, 2017 at 12:25 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
generalization to polynomial functions splitting on Q
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| Jul 11, 2017 at 1:27 | answer | added | Gerhard Paseman | timeline score: 6 | |
| Jul 11, 2017 at 0:43 | comment | added | Sebastien Palcoux | By searching the sequence $3,7,5,11,23,47,19,13$ on oeis, we find A287865 (posted on Jun 04 2017). It refers to Oskars Rieksts. He seems to know this problem since July 2016 (see here). I don't know if this problem was known to someone before him. | |
| Jul 11, 2017 at 0:27 | history | edited | Sebastien Palcoux | CC BY-SA 3.0 |
Minor edit
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| Jul 11, 2017 at 0:12 | history | asked | Sebastien Palcoux | CC BY-SA 3.0 |