{"id":3551,"date":"2023-03-09T15:22:39","date_gmt":"2023-03-09T20:22:39","guid":{"rendered":"https:\/\/people.clas.ufl.edu\/boyland\/?page_id=3551"},"modified":"2026-03-19T08:10:23","modified_gmt":"2026-03-19T12:10:23","slug":"sudhir-ghorpade","status":"publish","type":"page","link":"https:\/\/people.clas.ufl.edu\/boyland\/sudhir-ghorpade\/","title":{"rendered":"Sudhir Ghorpade"},"content":{"rendered":"\r\n<section class=\"fullwidth-text-block\">\r\n\t<div class=\"container px-0 pt-5\">\r\n\t\t<div class=\"row align-items-start\">\r\n\t\t\t<div class=\"col-12\">\r\n\t\t\t\t\n<h1 class=\"wp-block-heading\">Sudhir Ghorpade<\/h1>\n\n\n\n<h3 class=\"wp-block-heading\">Arithmetic Progressions in Unique Factorization Domains<\/h3>\n\n\n\n<p class=\"wp-block-paragraph\">Abstract: S. S. Pillai, a contemporary of Ramanujan and perhaps the second best Indian mathematician<br>\nat least of that era, proved in 1940 that any sequence of consecutive integers with at most 16 terms<br>\npossesses one term that is relatively prime to all the others. Later Brauer and Pillai showed (independently)<br>\nthat such a result is not true for sequences of length 17 or more. It is not difficult to see that Pillai&#8217;s theorem<br>\ngeneralizes from consecutive integers to arithmetic progressions in the sense that any sequence of 16 (or less)<br>\nintegers in a coprime arithmetic progression (that is, an a.p. for which the first term and common difference is<br>\nrelatively prime) necessarily contains a term that is relatively prime to all the others.<\/p>\n\n\n\n\n\n<p class=\"wp-block-paragraph\">We will discuss an algebraic extension of this Generalized Pillai Theorem in a significantly wider algebraic context.<br>\nThus, we ask, if a similar result holds for Gaussian integers, or more generally, for unique factorization domains,<br>\nor even more generally, for arbitrary integral domains where the notion of GCD (and hence of two elements<br>\nbeing relatively prime) makes sense. A nice example of such GCD domains is the ring of entire functions, thanks<br>\nto a result of Helmer, which was also published in 1940.<\/p>\n\n\n\n\n\n<p class=\"wp-block-paragraph\">We will outline a joint work with Samrith Ram that provides some answers to these questions.<\/p>\n\n\n\r\n\t\t\t<\/div>\r\n\t\t<\/div>\r\n\t<\/div>\r\n<\/section>\r\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":146,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_acf_changed":false,"featured_post":"","footnotes":"","_links_to":"","_links_to_target":""},"class_list":["post-3551","page","type-page","status-publish","hentry"],"acf":[],"_links":{"self":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/3551","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/users\/146"}],"replies":[{"embeddable":true,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/comments?post=3551"}],"version-history":[{"count":2,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/3551\/revisions"}],"predecessor-version":[{"id":3809,"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/pages\/3551\/revisions\/3809"}],"wp:attachment":[{"href":"https:\/\/people.clas.ufl.edu\/boyland\/wp-json\/wp\/v2\/media?parent=3551"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}