Timeline for A Collatz-like problem on prime numbers
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jul 25, 2017 at 19:47 | comment | added | Gottfried Helms | An additional bit of information can be found at keyword "Cunningham chain" ? en.wikipedia.org/wiki/Cunningham_chain There is also Aaron's number $p_1=2759...19$ mentioned as record holder found in 2008 | |
| Jul 11, 2017 at 15:38 | comment | added | Yaakov Baruch | @GerhardPaseman: you are correct about orbiting through 0. If $p$ is the starting prime (and $p$ doesn't divide the parameters of the affine transformation) then (Mod $p$) it will cycle trough 0 within $p-1$ steps. (A leisurely 2759832934171386593518 steps in the above example...) | |
| Jul 11, 2017 at 15:20 | history | edited | Gerhard Paseman | CC BY-SA 3.0 |
added 2025 characters in body
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| Jul 11, 2017 at 15:12 | history | edited | Gerhard Paseman | CC BY-SA 3.0 |
added 2025 characters in body
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| Jul 11, 2017 at 14:13 | comment | added | Aaron Meyerowitz | Easiest to imagine: it never happens but we will never see a proof. Much harder: a proof will be found. Hardest: an unbounded chain. | |
| Jul 11, 2017 at 13:25 | comment | added | Yaakov Baruch | For large numbers the expected drop at each down step is about 37% logarithmically, see Golomb-Dickman constant (this is not affected by the zero probability case of the number being prime and the drop being replaced by a logarithmically insignificant constant factor increase). So it's hard to imagine a path to infinity. | |
| Jul 11, 2017 at 13:05 | comment | added | Yaakov Baruch | @AaronMeyerowitz: alas, the 18th term is 361736822347711983585853439 with largest factor 55442017698213695587 and from there things quickly collapse into the known cycle. (This is not surprising as the larger the number the more likely it is to have factors, and even many large factors.) | |
| Jul 11, 2017 at 6:42 | comment | added | Gerhard Paseman | Hmm. I was pretty sure each affine map had a prime p in which the map not only had a finite period mod p but also had orbit through 0. Thanks for checking. I will sleep on this and hope for a better resolution. Gerhard "Maybe I Need More Citations" Paseman, 2017.07.10. | |
| Jul 11, 2017 at 6:32 | comment | added | Aaron Meyerowitz | Evidentally the map sending $p$ to $2p+1$ generates $17$ primes in a row if started at $p_1=2759832934171386593519.$ I didn't check but one might imagine that at the $18$th iteration the composition result (over $250000p_1$ has largest prime factor quite large and the start of another long chain. So even if there was s bound on how long it would take to get to a composite , it wouldn't rule out growth without bound. And the expected result is that there are arbitrarily long increasing chains. But I do agree that one would anticipate that everything does get to that cycle. | |
| Jul 11, 2017 at 1:27 | history | answered | Gerhard Paseman | CC BY-SA 3.0 |