Timeline for Does listing the prime factors always stop?
Current License: CC BY-SA 2.5
5 events
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| Jul 12, 2010 at 1:34 | comment | added | Wadim Zudilin | Thanks, Junkie! It's indeed a vague argument but it can be developed into some kind of rigorous heuristic. | |
| Jul 12, 2010 at 0:55 | comment | added | Junkie | Here is the vague heuristic. Factoring into $r$ primes appends about $r$ digits. The typical number of prime factors of a number of size $N$ (or $\log N$ digits) is $\log\log N$. The idea is that if we start with a number of $k$ digits, it will be typically mapped to one with about $k+\log k$ digits. Iterating the size function in digits is around $F(t)=k+t\log kt$ at the $t$th iterate, as the $t$ dominates. The chance of a number with $D$ digits being prime is about $1/D$. Summing $1/F(t)$ over $t$ diverges. This is rough and I ignore constants through it. | |
| Jul 12, 2010 at 0:17 | comment | added | Wadim Zudilin | Charles, as far as I understand Conway's heuristic, you are right. I was trying to produce an evidence to myself for the fact that any sequence eventually stops. 77 and 300 do not (experimentally, of course), so I indicated my doubts. In any case, the problem looks less treatable than $3x+1$, because the corresponding dynamics can be hardly described. | |
| Jul 11, 2010 at 23:10 | comment | added | Charles | Let's not duplicate effort: factordb.com/… I disagree on your heuristic, though. The numbers get larger so the chance of a particular number being prime decreases, but the chance of getting one eventually seems quite good. Of course for many examples it will be far beyond what we can calculate... | |
| Jul 11, 2010 at 14:02 | history | answered | Wadim Zudilin | CC BY-SA 2.5 |