Consider a sequence $(S_i)_{i=1}^{\infty},\ S_i=\pm1$ other than $\{1,1,\ldots\}$. Let $g_i=p_{i+1}-p_i$, where $p_i$ is the $i$th prime. Is it possible that for all $k\in\Bbb Z^+,\ 2+\sum_{i=1}^kS_ig_i\in\Bbb P$?
Based on limited numerical evidence, the answer appears negative, as the number of such sequences of length $k$ frequently collapses to $1$ and never seems to grow very large. Surprisingly, when the primes are replaced with practical numbers (and $2$ replaced with $1$, their initial term), which grow at a similar rate and also have a "random" nature to their distribution, the answer seems to be positive, with the number of such sequences greater than $1$ for $k>2$ as far as I've seen. I'm not so interested in addressing this question for practical numbers, as it seems that increasing sequences of positive integers with this property should be fairly common, but I wonder if there are any significant implications of a sequence not having this property. In particular, whether this says anything established for or conjectured of primes.
Yes. Consider the sequence $s$ of $1$'s and $-1$'s defined by \begin{eqnarray} s_i := \begin{cases}-1, &\mbox{ if }i = 3,\\+1, &\mbox{ otherwise,}\end{cases} \end{eqnarray} and note that $q_k := 2 + \sum_{1 \le i \le k}s_ig_i = \begin{cases}3, &\mbox{ if }k=1,\\5, &\mbox{ if }k=2,\\3 &\mbox { if }k=3,\\p_k, &\mbox{ if }k \ge 4,\end{cases}$
which is essentially the entire sequence of primes! I think you'll need to put more restrictions on the sequence $s$ to get more interesting "walks" on the primes :)