Ah! Good point. Jim On Tuesday, June 19, 2018, Michael Kleber <michael.kleber@gmail.com> wrote:
Well certainly x^2-1 can never go critical because, because it can never even increase! If p is prime, then p^2-1 = (p+1)(p-1), both of which must be composite themselves (once p>3), so you never create a prime larger than the one you started with.
--Michael
On Tue, Jun 19, 2018 at 8:54 AM James Propp <jamespropp@gmail.com> wrote:
I did a few experiments with x^2-1 in my head, and in every case, the reaction was subcritical.
Is it possible that x^2-1 is subcritical for every initial seed, whereas x^2+1 Is supercritical for every non-empty initial seed?
Jim
On Monday, June 18, 2018, James Propp <jamespropp@gmail.com> wrote:
What happens with x^2-1?
Jim Propp
On Monday, June 18, 2018, <rcs@xmission.com> wrote:
Here's a related result: You can't have everything except a finite number of missing primes. In fact, you can't have 100% - epsilon, with epsilon->0.
Why not? Suppose 17 is missing. Since sqrt(-1) = +-4 (mod 17), all the primes 17k+-4 must also be missing. This mean 2/16 = 12.5% (at least) are missing. This contradicts the 100%-epsilon hypothesis. [QED]
Moreover, these missing primes in turn create another set of missing primes. This suggests that the only possibilities for the final set are All Primes, or some much smaller percentage, maybe 0%. This would still permit an infinite number of primes, but they'd need to be sparse.
Notice that F(x) = x^2 - 1 behaves very differently.
I did a couple of experiments with simply iterating F(x) to make a sequence, rather than building out the whole tree. F(x) = largest-prime-factor(x^2+1) seems to head for infinity. The ride is bumpy, but I'm up to 70 digit numbers. I did a small experiment with F(x) = LPF(floor(x^1.5)); all x<101 peter out into small numbers, although I did get some intermediate 10 & 15-digit values. 157-281-157 is a small loop.
Rich
---- Quoting James Propp <jamespropp@gmail.com>:
With the polynomial F(x) = x^2 + 1, the seed 1 appears to yield all the primes it for which -1 is a quadratic residue, i.e., the prime 2 and all primes congruent to 1 mod 4. (Note that these are the only primes we could hope to get.)
Can anyone prove this?
I've obtained all the 1-mod-4 primes up to 1000.
Jim
On Sunday, June 17, 2018, James Propp <jamespropp@gmail.com> wrote:
My intuition is that F(x) = x^2 + 1 is supercritical.
Jim
On Sunday, June 17, 2018, Warren D Smith <warren.wds@gmail.com>
wrote:
If in your process instead of doubling & add 1, i.e. the map 2x+1, > do the map "F(x)" for integer polynomials F > I would guess for fast enough growing F(x) > the process ought to blow > to create an infinite set of primes while for slow enough F it will > not. > > I.e. I suspect there is a "critical mass" phenomenon where at some > point > you are breeding new primes fast enough to create exponential > population > explosion, but below that point it self-limits. > > So what sort of growth for F constitutes that "critical mass"? > Interesting & likely delicate question. > > Just as an initial guess, perhaps > F(X) = 1 + X^floor(lnlnX) > is supercritical, but F = any polynomial(X) is subcritical. > > -- > Warren D. Smith > http://RangeVoting.org <-- add your endorsement (by clicking > "endorse" as 1st step) > > _______________________________________________
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