2013/12/26 by Romeo Meštrović, Meštrović, Romeo
Engineering · Mathematics · #11A05 #11A07 #11B73 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Arithmetic #Combinatorics #Congruence (geometry) #Congruence relation #Counterexample #Discrete mathematics #Divisibility rule #Divisor (algebraic geometry) #FOS: Mathematics #Factorial #Generalization #Integer (computer science) #Mathematics #Multiple #Number Theory (math.NT) #Number theory #Order (exchange) #Primary 05A10 #Prime (order theory) #Prime factor #Secondary 11B65 #graph theory and CDMA systems #math.NT #msc:05A10 #msc:11A05 #msc:11A07 #msc:11B65 #msc:11B73
paper · pdf · doi:10.48550/arxiv.1312.7037
18 pages. This is the previous (first) version of the article extended with Section 4 where we disprove the "odd composite part''of Strong Kurepa's hypothesis
openalex publication_date 2013/12/26 · arxiv created 2014/01/06 · arxiv updated 2014/01/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Kurepa's hypothesis asserts that for each integer n≥ 2 the greatest common divisor of !n:=∑k=0n-1k! and n! is 2. Motivated by an equivalent formulation of this hypothesis involving derangement numbers, here we give a formulation of Kurepa's hypothesis in terms of divisibility of any Kurepa's determinant Kp of order p-4 by a prime p≥ 7. In the previous version of this article we have proposed the strong Kurepa's hypothesis involving a general Kurepa's determinant Kn with any integer n≥ 7. We prove the ``even part'' of this hypothesis which can be considered as a generalization of Kurepa's hypothesis. However, by using a congruence for Kn involving the derangement number Sn-1 with an odd integer n≥ 9, we find that the integer 11563=31× 373 is a counterexample to the ``odd composite part'' of strong Kurepa's hypothesis. We also present some remarks, divisibility properties and computational results closely related to the questions on Kurepa's hypothesis involving derangement numbers and Bell numbers.