New Equivalents of Kurepa’s Hypothesis for Left Factorial
Author:
Petojević Aleksandar1,
Gordić Snežana1ORCID,
Mandić Milinko1,
Ranitović Marijana Gorjanac1ORCID
Affiliation:
1. Faculty of Education in Sombor, University of Novi Sad, Podgorička 4, 25000 Sombor, Serbia
Abstract
Kurepa’s hypothesis for the left factorial has been an unsolved problem for more than 50 years. In this paper, we have proposed new equivalents for Kurepa’s hypothesis for the left factorial. The connection between the left factorial and the continued fractions is given. The new equivalent based on the properties of the integer part of real numbers is proven. Moreover, a new equivalent based on the properties of two well-known sequences is given. A new representation of the left factorial is listed. Since derangement numbers are closely related to Kurepa’s hypothesis, we made some notes about the derangement numbers and defined a new sequence of natural numbers based on the derangement numbers. In this paper, we indicate a possible direction for further research through solving quadratic equations.
Funder
Ministry of Sciences and Technological Development of the Republic of Serbia through the Faculty of Education, University of Novi Sad
Subject
Geometry and Topology,Logic,Mathematical Physics,Algebra and Number Theory,Analysis
Reference53 articles.
1. Weil, A. (1984). Number Theory: An Approach through History from Hammurapi to Legendre, Springer.
2. On the left factorial function !n;Kurepa;Math. Balk.,1971
3. Guy, R. (1981). Unsolved Problems in Number Theory, Springer.
4. Koninck, J.M.D., and Mercier, A. (2007). 1001 Problems in Classical Number Theory, American Mathematical Society.
5. Sándor, J., and Crstici, B. (2004). Handbook of Number Theory II, Kluwer.