Parallel Searching Method for Verifying Finiteness of Prime Numbers Presented as Specific Linear Combinations of Factorials

As a new approach to search a counterexample of variants of Kurepa’s conjecture, we investigate a parallel searching method of prime numbers $p$ satisfying \begin{eqnarray*} \sum_{k=0}^{p-1} c_k k! \equiv 0 \pmod{p} \end{eqnarray*} for a given integer sequence $(c_k{)}_{k=0}^{\mathrm{\infty }}$. As an application, we prove that there are only finitely may prime numbers of the form ${\sum }_{k=0}^n(1+dk)!$ with $(d,n)\in ({\mathbb{N}}_{\le 25}\setminus \{0,8,18,20,23,25\})\times \mathbb{N}$.