Even easier would be f(x) = x+2 and K=1, we get all odd integers and they do contain infinitely many primes. Warm regards, Shripad. On Tue, Jun 4, 2013 at 10:52 AM, Fred W. Helenius <fredh@ix.netcom.com>wrote:
On 6/4/2013 1:05 AM, Dan Asimov wrote:
What if any is the current thinking on this question:
Does there exist any nontrivial integer polynomial
f(x) := A_n x^n + . . . + A_0
with n > 0 and A_n > 0, such that for some positive integer K the sequence
K, f(K), f(f(K)), f(f(f(K))), . . .
contains infinitely many primes?
What do you mean by "nontrivial"? f(x) = x + b will work for any positive b and any K coprime to b (by Dirichlet's theorem).
If f(x) = 2x + 1 and K = 1, then the resulting sequence contains the Mersenne numbers. It is widely believed that there are infinitely many Mersenne primes.
-- Fred W. Helenius fredh@ix.netcom.com
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-- Shripad M. Garge. / श्रीपाद म. गर्गे. Department of Mathematics, Indian Institute of Technology, Powai, Mumbai 400 076. 022-2576-7473/8473 (Off./Res.) http://www.math.iitb.ac.in/~shripad/