If we start with a nonempty collection of odd primes, x^2+2 seems to give all the primes that are 1 or 3 mod 8, whereas x^2-2 seems to give all the primes that are 1 or 7 mod 8. These are all the odd primes we could hope for. Jim Propp On Tue, Jun 19, 2018 at 10:43 AM, Lucas, Stephen K - lucassk < lucassk@jmu.edu> wrote:
So instead of x^2+1, which will be composite when x is prime, how about x^2+2?
Steve
On Jun 17, 2018, at 7:08 PM, W. Edwin Clark <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> 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.
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