Parallel Searching Method for Verifying Finiteness of Prime Numbers Presented as Specific Linear Combinations of Factorials
Year: 2026
Type: preprint
Abstract: 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 $(c_k)_{k=0}^{\infty}$. As an application, we prove that there are only finitely may prime numbers of the form $\sum_{k=0}^{n} (1+dk)!$ with $(d,n) \in (\mathbb{N}_{\leq 25} \setminus \{0,8,18,20,23,25\}) \times \mathbb{N}$.
Source: arXiv (Cornell University)
Author: Mihara, Tomoki
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214384085