{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:00:03Z","timestamp":1760148003529,"version":"build-2065373602"},"reference-count":14,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,3,24]],"date-time":"2023-03-24T00:00:00Z","timestamp":1679616000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Transilvania University of Bra\u015fov"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this article, we find some properties of certain types of entropies of a natural number. We are studying a way of measuring the \u201cdisorder\u201d of the divisors of a natural number. We compare two of the entropies H and H\u00af defined for a natural number. An useful property of the Shannon entropy is the additivity, HS(pq)=HS(p)+HS(q), where pq denotes tensor product, so we focus on its study in the case of numbers and ideals. We mention that only one of the two entropy functions discussed in this paper satisfies additivity, whereas the other does not. In addition, regarding the entropy H of a natural number, we generalize this notion for ideals, and we find some of its properties.<\/jats:p>","DOI":"10.3390\/e25040554","type":"journal-article","created":{"date-parts":[[2023,3,24]],"date-time":"2023-03-24T03:16:46Z","timestamp":1679627806000},"page":"554","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["About the Entropy of a Natural Number and a Type of the Entropy of an Ideal"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0716-293X","authenticated-orcid":false,"given":"Nicu\u015for","family":"Minculete","sequence":"first","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Street 50, 500091 Bra\u015fov, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Diana","family":"Savin","sequence":"additional","affiliation":[{"name":"Faculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Street 50, 500091 Bra\u015fov, Romania"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"110360","DOI":"10.1016\/j.chaos.2020.110360","article-title":"New entropy bounds via uniformly convex functions","volume":"141","author":"Sayyari","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_2","first-page":"1112797","article-title":"An improved estimator of Shannon entropy with applications to systems with memory","volume":"165","author":"Toral","year":"2022","journal-title":"Chaos Solitons Fractals"},{"key":"ref_3","first-page":"205","article-title":"Algebraic entropies of natural numbers with one or two factors","volume":"23","author":"Jeong","year":"2016","journal-title":"J. Korean Soc. Math. Educ. Ser. B Pure Appl. Math."},{"key":"ref_4","unstructured":"(2022, August 01). Available online: https:\/\/math.stackexchange.com\/questions\/2369779\/entropy-of-a-natural-number."},{"key":"ref_5","first-page":"425","article-title":"The Entropy of a Natural Number","volume":"4","author":"Minculete","year":"2011","journal-title":"Acta Tech. Jaurinensis"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"344","DOI":"10.1016\/j.exmath.2020.07.001","article-title":"Some generalizations of the functions \u03c4 and \u03c4(e) in algebraic number fields","volume":"39","author":"Minculete","year":"2021","journal-title":"Expo. Math."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Furuichi, S., and Minculete, N. (2021). Refined Young Inequality and Its Application to Divergences. Entropy, 23.","DOI":"10.3390\/e23050514"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1442","DOI":"10.1103\/PhysRevE.58.1442","article-title":"Generalized entropy-based criterion for consistent testing","volume":"58","author":"Tsallis","year":"1998","journal-title":"Phys. Rev. E"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Niepostyn, S.J., and Daszczuk, W.B. (2023). Entropy as a Measure of Consistency in Software Architecture. Entropy, 25.","DOI":"10.3390\/e25020328"},{"key":"ref_10","unstructured":"Ireland, K., and Rosen, M. (1992). A Classical Introduction to Modern Number Theory, Springer."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ribenboim, P. (2000). My Numbers, My Friends (Popular Lectures on Number Theory), Springer.","DOI":"10.1007\/b98892"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Ribenboim, P. (2001). Classical Theory of Algebraic Numbers, Springer.","DOI":"10.1007\/978-0-387-21690-4"},{"key":"ref_13","unstructured":"Savin, D., and \u015etefanescu, M. (2008). Lessons of Arithmetics and Number Theory, Matrix Rom Publishing House. (In Romanian)."},{"key":"ref_14","unstructured":"Murty, M.R., and Esmonde, J. (2005). Problems in Algebraic Number Theory, Springer. [2nd ed.]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/4\/554\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:01:55Z","timestamp":1760122915000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/4\/554"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,3,24]]},"references-count":14,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4]]}},"alternative-id":["e25040554"],"URL":"https:\/\/doi.org\/10.3390\/e25040554","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2023,3,24]]}}}