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

If the degree is >1, and ignoring a couple of trivial looping cases, the numbers will grow roughly as C^N^J for J=1,2,3,.... and some C>1. If we adopt the "primes are random with probability 1/logX" model, then the expected number of primes is C' sum (1/logC) N^-J (C' is a correction factor reflecting that the sequence may consist of only odd numbers etc.) This series converges, so the expected number of primes is finite; in fact, rather small. This seems to be the problem with Fermat primes, as an example; the iteration polynomial is Xnew = (Xold-1)^2 + 1. Someone has pointed out that the Chudnovsky paper about primes in elliptic-curve related sequences generates numbers around C^J^2. As do many Somos sequences. The expected number of primes is C' sum (1/logC) J^-2 which is also a convergent series. So, the simple primes-are-random model predicts only finitely many primes in these sequences. Sometimes a sequence will have an approximate sin(CJ) multiplier, which can occasionally reduce the size of particular sequence values. This depends on the continued fraction of pi/C, making it possible that small values with higher prime chances persist indefinitely. This won't happen with Dan's polynomial examples, but can occur in Somos-ish and elliptic-curve situations. Rich ________________________________________ From: math-fun-bounces@mailman.xmission.com [math-fun-bounces@mailman.xmission.com] on behalf of Dan Asimov [dasimov@earthlink.net] Sent: Monday, June 03, 2013 11:05 PM To: math-fun Subject: [EXTERNAL] [math-fun] Primes question 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? --Dan _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun