{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T21:45:21Z","timestamp":1740174321601,"version":"3.37.3"},"reference-count":4,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T00:00:00Z","timestamp":1689552000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T00:00:00Z","timestamp":1689552000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e Tec"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Bull. Malays. Math. Sci. Soc."],"published-print":{"date-parts":[[2023,9]]},"abstract":"Abstract<\/jats:title>We give necessary and sufficient conditions for the divisibility of two finite geometric series $$ G_n(x) = 1 + x + x ^ 2 + \\cdots + x^{n-1} $$<\/jats:tex-math>\n \n \n G<\/mml:mi>\n n<\/mml:mi>\n <\/mml:msub>\n \n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n 1<\/mml:mn>\n +<\/mml:mo>\n x<\/mml:mi>\n +<\/mml:mo>\n \n x<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n \u22ef<\/mml:mo>\n +<\/mml:mo>\n \n x<\/mml:mi>\n \n n<\/mml:mi>\n -<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math><\/jats:alternatives><\/jats:inline-formula> over a field of characteristic zero.<\/jats:p>","DOI":"10.1007\/s40840-023-01556-5","type":"journal-article","created":{"date-parts":[[2023,7,17]],"date-time":"2023-07-17T14:02:06Z","timestamp":1689602526000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Divisibility of Finite Geometric Series"],"prefix":"10.1007","volume":"46","author":[{"given":"Robert E.","family":"Hartwig","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9776-0919","authenticated-orcid":false,"given":"Pedro","family":"Patr\u00edcio","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,7,17]]},"reference":[{"key":"1556_CR1","doi-asserted-by":"crossref","unstructured":"Bac, D.H., Binh, N., Quynh, N.X.: New algebraic structure based on cyclic geometric progressions over polynomial ring applied for cryptography. In: Proceedings\u2014CIS Workshops 2007, International Conference on Computational Intelligence and Security Workshops, art. no. 4425610, pp. 777\u2013780 (2007)","DOI":"10.1109\/CISW.2007.4425610"},{"issue":"4","key":"1556_CR2","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1007\/s00013-020-01488-7","volume":"115","author":"MB Nathanson","year":"2020","unstructured":"Nathanson, M.B.: Geometric progressions in syndetic sets. Archiv der Mathematik 115(4), 413\u2013417 (2020)","journal-title":"Archiv der Mathematik"},{"issue":"2","key":"1556_CR3","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1007\/s00013-019-01320-x","volume":"113","author":"BR Patil","year":"2019","unstructured":"Patil, B.R.: Geometric progressions in syndetic sets. Archiv der Mathematik 113(2), 157\u2013168 (2019)","journal-title":"Archiv der Mathematik"},{"issue":"14","key":"1556_CR4","doi-asserted-by":"publisher","first-page":"4613","DOI":"10.2298\/FIL2114613P","volume":"35","author":"P Patr\u00edcio","year":"2021","unstructured":"Patr\u00edcio, P., Hartwig, R.E.: From euclid to corner sums, a trail of telescoping tricks. Filomat 35(14), 4613\u20134636 (2021)","journal-title":"Filomat"}],"container-title":["Bulletin of the Malaysian Mathematical Sciences Society"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40840-023-01556-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s40840-023-01556-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s40840-023-01556-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,9,11]],"date-time":"2023-09-11T12:14:01Z","timestamp":1694434441000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s40840-023-01556-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,17]]},"references-count":4,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["1556"],"URL":"https:\/\/doi.org\/10.1007\/s40840-023-01556-5","relation":{},"ISSN":["0126-6705","2180-4206"],"issn-type":[{"type":"print","value":"0126-6705"},{"type":"electronic","value":"2180-4206"}],"subject":[],"published":{"date-parts":[[2023,7,17]]},"assertion":[{"value":"26 May 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 June 2023","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 July 2023","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 July 2023","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors have no conflicts of interest to declare.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"161"}}