{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:25:16Z","timestamp":1760149516949,"version":"build-2065373602"},"reference-count":53,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,12]],"date-time":"2023-08-12T00:00:00Z","timestamp":1691798400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Sciences and Technological Development of the Republic of Serbia through the Faculty of Education, University of Novi Sad"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Kurepa\u2019s 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\u2019s 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\u2019s 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.<\/jats:p>","DOI":"10.3390\/axioms12080785","type":"journal-article","created":{"date-parts":[[2023,8,14]],"date-time":"2023-08-14T11:07:10Z","timestamp":1692011230000},"page":"785","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["New Equivalents of Kurepa\u2019s Hypothesis for Left Factorial"],"prefix":"10.3390","volume":"12","author":[{"given":"Aleksandar","family":"Petojevi\u0107","sequence":"first","affiliation":[{"name":"Faculty of Education in Sombor, University of Novi Sad, Podgori\u010dka 4, 25000 Sombor, Serbia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8039-0456","authenticated-orcid":false,"given":"Sne\u017eana","family":"Gordi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Education in Sombor, University of Novi Sad, Podgori\u010dka 4, 25000 Sombor, Serbia"}]},{"given":"Milinko","family":"Mandi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Education in Sombor, University of Novi Sad, Podgori\u010dka 4, 25000 Sombor, Serbia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6811-4993","authenticated-orcid":false,"given":"Marijana Gorjanac","family":"Ranitovi\u0107","sequence":"additional","affiliation":[{"name":"Faculty of Education in Sombor, University of Novi Sad, Podgori\u010dka 4, 25000 Sombor, Serbia"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,12]]},"reference":[{"key":"ref_1","unstructured":"Weil, A. 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