{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T03:03:18Z","timestamp":1760151798913,"version":"build-2065373602"},"reference-count":3,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,4,19]],"date-time":"2022-04-19T00:00:00Z","timestamp":1650326400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A positive integer, which can be written as the sum of two positive cubes in two different ways, is known as a \u201cRamanujan number\u201d. The most famous example is 1729=103+93=123+13, which was identified by Ramanujan as the lowest such number. In this paper, we consider the homogeneous cubic Diophantine equation x3+y3=u3+v3, where there is no restriction on the signs of the integers x,y,u,v. We show that every solution can be written in terms of two parameters in the ring Z\u22123. It is also shown that solutions with arbitrarily high values of max(|x|,|y|,|u|,|v|) arise amongst the primitive solutions.<\/jats:p>","DOI":"10.3390\/axioms11050184","type":"journal-article","created":{"date-parts":[[2022,4,19]],"date-time":"2022-04-19T22:07:26Z","timestamp":1650406046000},"page":"184","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A New Solution to a Cubic Diophantine Equation"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6499-9851","authenticated-orcid":false,"given":"John","family":"Butcher","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Auckland, Auckland 1010, New Zealand"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,4,19]]},"reference":[{"key":"ref_1","unstructured":"Hardy, G.H. (1940). A Mathematician\u2019s Apology, Cambridge University Press."},{"key":"ref_2","first-page":"21","article-title":"On a Cubic Diophantine Equation","volume":"427","author":"Goormaghtigh","year":"1937","journal-title":"Math. Gazette"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1080\/00029890.1993.11990409","article-title":"Taxicabs and sums of two cubes","volume":"100","author":"Silverman","year":"1993","journal-title":"Am. Math. Mon."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/5\/184\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T22:56:32Z","timestamp":1760136992000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/5\/184"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,4,19]]},"references-count":3,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["axioms11050184"],"URL":"https:\/\/doi.org\/10.3390\/axioms11050184","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,4,19]]}}}