Denote $p_j := j\text{th prime}$ and $S(n)\:=\sum_{j=1}^n p_j$ (The sum of the first $n$ primes).

Is it known whether $S(n)$ is prime for infinite many $n$?

OEIS gives the sum of the prime numbers upto $10^{18}$, which is $12,212,914,292,949,226,570,880,576,733,896,687$, being a $35$-digit prime.

The relative low growth rate of $S(n)$ (of order $n^2\ln n$) is another heuristic argument that the answer to the given question should be yes.