What is the longest known sequence of primes where each new term is obtained by appending a new decimal digit to the previous term?
There are no more members in this sequence, because $233331,233333,233337,233339$ are all composite numbes.
This one is longer, but still only $8$ terms.
An so on. Because prime gaps can become arbitrary long for large numbers, there is no way any such sequence will be infinite. On the other hand, they can be very long, I think.
What is the longest known sequence of this kind?
Can we prove that:
- There is no infinite sequence of this kind. Or at least that the sum of reciprocals of any such sequence is finite.
Of course, we can study such sequences in binary or other bases.
The sequence can start with any prime. This link is very relevant. See also the comments.
The argument that an infinite sequence like this is unlikely goes like this: For each new term we have only four possibilities:
All of these numbers are contained in a very small interval, about $10$ in size. But the density of primes becomes smaller with growing number of digits, and the gaps become larger. In my opinion, the probability of finding the next term for any such sequence becomes smaller with growing $n$.
It doesn't mean that an infinite sequence is impossible, but I conjecture that it's impossible to prove that any such infinite sequence exists.