Erdős Problem 427
Statement.
For positive integers $n,d$, does some $k\geq1$ satisfy \[
d\mid p_{n+1}+\cdots+p_{n+k},
\] where $p_j$ is the $j$-th prime?
External input.
We use Shiu's theorem on strings of congruent primes: if $q\geq3$ and $\gcd(a,q)=1$, there exist arbitrarily long strings of consecutive primes all congruent to $a\pmod q$. This is the deep input; its proof is not reproduced here. It implies that a string of any prescribed finite length exists beyond any prescribed index: otherwise strings of unbounded length would have bounded starting index and would force all sufficiently late primes to lie in that residue class, which also supplies arbitrarily late strings.
Deduction of the requested divisibility.
The case $d=1$ is immediate. For $d=2$, the primes after $p_n$ are odd because $n\geq1$; two of them have an even sum. For $d\geq3$, apply Shiu's theorem with $a=1$, $q=d$. Choose $m>n$ such that \[
p_m,\ldots,p_{m+d-1}\equiv1\pmod d.
\] Write $s=\sum_{j=n+1}^{m-1}p_j$, allowing an empty sum. There is a unique $\ell\in\{1,\ldots,d\}$ with $\ell\equiv-s\pmod d$. Then \[
\sum_{j=n+1}^{m+\ell-1}p_j
\equiv s+\ell\equiv0\pmod d.
\] Taking $k=m+\ell-1-n\geq1$ proves the assertion. Choosing strings arbitrarily far out gives infinitely many such $k$ for each fixed $n,d$.
Attribution and scope.
This reconstructs the deduction attributed to Cedric Pilatte on the problem page. Its external input is D. K. L. Shiu,
Strings of congruent primes, Journal of the London Mathematical Society 61 $2000$, 359--373,
https://doi.org/10.1112/S0024610799007863. The current statement is
https://www.erdosproblems.com/427, checked 24 September 2026. Completeness is relative to the explicitly stated established theorem, not an independent proof of Shiu's theorem.
Completion estimate.
The full assertion follows from Shiu's theorem with all quantifiers checked.
COMPLETION: 100%