----- Original Message ----- From: <asimovd@aol.com> To: <math-fun@mailman.xmission.com> Sent: Friday, February 28, 2003 2:18 AM Subject: [math-fun] OOPS! Re: (Another) Prime Question

As stated, my question, except for p = 2, is nonsense, since multiplication by p and then adding one takes an odd number to an even one..

So -- what if we just pretend I never mentioned any prime but 2:

<< Consider the mapping f: N -> N given by f(n) = 2n + 1.

Consider all infinite sequences

(*) q_0, q_1, q_2, ..., where q_(n+1) = f(q_n) for n = 0,1,2,...

1. Is there an infinite prime sequence (*) ?

No. Choose prime m so that some q(n) == 0 (mod m). The sequence of residues {q(n) mod m} are cyclic in n, so some q(w) == 0 (mod m) with w > n ==> q(w) > q(n). This would make q(w) a composite with divisor m.

2. If not, are the lengths of such sequences unbounded? a) in any one sequence?

No. There will be no more than 2m consecutive primes, where m is the smallest prime divisor of any a(n).

b) if not, then perhaps among all possible such sequences?

In theory, yes.

3. If not, what's the length of the longest sequence of primes? a) in one sequence b) among all such sequences?

The longest run of primes among all such sequences (if it exists) would be the longest run of primes in the particular sequence in which it occurs, so (a) and (b) are the same question. We don't know the answer to this question. We don't even know if there is an arbitrarily long run of primes in arithmetic progression (your problem with f(n) = n + c), though we suspect there is.

4. Is there even a sequence containing infinitely many primes?

I suspect we don't know. We certainly don't know for the sequence with q(0) = 1, which is the sequence of Mersenne numbers. In this case, we keep finding Mersenne primes, but we don't know if there are infinitely many of them, we like to think so.

5. If so, what's the least upper bound of the various densities of the primes in such sequences, and is it attained?

Again, very difficult stuff.

--Dan A.