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 (*) ? 2. If not, are the lengths of such sequences unbounded? a) in any one sequence? b) if not, then perhaps among all possible such sequences? 3. If not, what's the length of the longest sequence of primes? a) in one sequence b) among all such sequences? 4. Is there even a sequence containing infinitely many primes? 5. If so, what's the least upper bound of the various densities of the primes in such sequences, and is it attained?
--Dan A.