Given a base, $b$ is there is a sequence $\lbrace a_n\rbrace_{n\geq 0}$ where $a_k \in \lbrace 1,2\cdots, b-1\rbrace$so that the sequence:
$$b_n:= \sum_{k=0}^n a_kb^k$$
is a sequence of primes numbers (or $1$, allowing $a_0$ to be be $1$)?
For $b=2$ the sequence is finite: $1,11,111$ are primes (or $1$) but $1111 = 11·101$
For $b=3$ the longest of such sequences of primes is $2,12,212$ but $1212 = 101·12$ and $2212 = 102·21$
I've noticed that $1,31,331,3331,\cdots,33333331$ in base $10$, are primes thus far but I haven't been able to check if it's true for $333333331$ or $a33333331$ for that matter (also, it doesn't apear in OEIS). But I'm almost certain that eventually that sequence will lead to a composite number.
So is there a base $b$ so that the sequence of primes is infinite?
For the sequence $1,31,331,\cdots$ what's the smallest composite?