Going back a ways in this discussion, the map 3x+2 produces a different set from 2x+1, but seems to have the same sort of behavior. Starting with 7, you get 5 7 13 17 23 41 43 53 71 79 83 127 131 151 239 251 383 691 719 1151 and starting with 59 you get 5 7 11 13 17 23 41 43 53 59 71 79 83 127 131 151 179 239 251 383 691 719 1151 and so on. Here is a terrible probabilistic argument for why this sort of process will always be finite. Primes become increasingly sparse as n increases, so the probability that 2x+1 is prime goes down as x increases, so you have to be lucky to get a very large number that is also prime. Eventually, the run of chance means you won’t be able to go any higher. Yes, I know numbers are prime or not regardless of probabilities related to the prime number theorem, but it does seem back of the envelope reasonable. Steve On Jun 17, 2018, at 7:08 PM, W. Edwin Clark <wclark@mail.usf.edu<mailto:wclark@mail.usf.edu>> wrote: if x is an odd prime and k is a positive integer then x^k+1 is not prime. On Sun, Jun 17, 2018 at 5:59 PM, James Propp <jamespropp@gmail.com<mailto: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<mailto: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 https://urldefense.proofpoint.com/v2/url?u=http-3A__RangeVoting.org&d=DwICAg... <-- add your endorsement (by clicking "endorse" as 1st step) _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com<mailto:math-fun@mailman.xmission.com> https://urldefense.proofpoint.com/v2/url?u=https-3A__mailman.xmission.com_cg...