Erdős Problem 427: is it true that, for every $n$ and $d$, there exists $k$ such that $$d \mid p_{n + 1} + \cdots + p_{n + k},$$ where $p_r$ denotes the $r$th prime?

Shiu's theorem: for any $k \geq 1$ and $(a, q) = 1$ there exist infinitely many $k$-tuples of consecutive primes $p_m, \dots, p_{m + k - 1}$ all of which are congruent to $a$ modulo $q$.

[Sh00] Shiu, D. K. L., _Strings of congruent primes_. J. London Math. Soc. (2) (2000), 359-373.