Erdős Problem #427
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
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The predicate 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?
The statement of 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.
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