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Let $P(x)$ be the xth prime, $P(0)=2$

Given a prime $p$, does it always exists a $n$ such $\sum_{x=0}^{n}P(x) = 0$ $(mod$ $p)$ ?

Example :

$p=7$

$2+3+5+7+11=28=4*7$

Best regards

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    i have an another question : is there well known universal formula for generation of prime numbers? if not than this question seems unanswerable – 2017-01-02

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http://oeis.org/A111287 is smallest $k$ such that $p_n$ divides $\sum_{i=1}^kp_i$. It says, "It follows from a theorem of Daniel Shiu that $k$ always exists," and many more details are given. [And I'm pretty sure I have already seen, maybe even posted, this answer to m.se before]

EDIT: I found the earlier question, Every prime number divide some sum of the first $k$ primes. and have voted to close this one as a duplicate.