<<
First, what about other bases besides ten? [This is normally dan asimov's
question, isn't it?]
>>

I must express my gratitude for someone's finally taking the burden off me in this case (:-)>.

In fact I was thinking of related questions that aren't directly base-related.  Here's one:

For any prime p consider the mapping f_p : N -> N given by f_p(n) = p*n + 1.

Consider (possibly finite) sequences of primes

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

Given prime p fixed:
1.    Is there an infinite prime sequence (*) ?

2.    If not, are the lengths of such sequences unbounded?

3.    If not, what's the length of the longest sequence?

And: How do the answers to the above questions depend on p ?

(Variants: one could look at g_p(n) = p*n - 1, or allowing +-1 at each iteration, or even allow any of the f_p's (or g_p's) to be applied at any stage in the iteration.)

But to start with, I'm most interested in the original question for p = 2.

--Dan A.