{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:52:41Z","timestamp":1776840761265,"version":"3.51.2"},"reference-count":23,"publisher":"American Mathematical Society (AMS)","issue":"267","license":[{"start":{"date-parts":[[2010,1,30]],"date-time":"2010-01-30T00:00:00Z","timestamp":1264809600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n In this paper we construct a cover\n \n \n \n \n {<\/mml:mo>\n \n a<\/mml:mi>\n \n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n mod<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n \u00a0<\/mml:mtext>\n \n n<\/mml:mi>\n \n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n \n }<\/mml:mo>\n \n s<\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n \\{a_{s}(\\operatorname {mod} \\ n_{s})\\}_{s=1}^{k}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {Z}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with odd moduli such that there are distinct primes\n \n \n \n \n \n p<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n p<\/mml:mi>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n p_{1},\\ldots ,p_{k}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n dividing\n \n \n \n \n \n 2<\/mml:mn>\n \n \n n<\/mml:mi>\n \n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n 2<\/mml:mn>\n \n \n n<\/mml:mi>\n \n k<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 2^{n_{1}}-1,\\ldots ,2^{n_{k}}-1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n respectively. Using this cover we show that for any positive integer\n \n \n \n m<\/mml:mi>\n m<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n divisible by none of\n \n \n \n \n 3<\/mml:mn>\n ,<\/mml:mo>\n 5<\/mml:mn>\n ,<\/mml:mo>\n 7<\/mml:mn>\n ,<\/mml:mo>\n 11<\/mml:mn>\n ,<\/mml:mo>\n 13<\/mml:mn>\n <\/mml:mrow>\n 3, 5, 7, 11, 13<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n there exists an infinite arithmetic progression of positive odd integers the\n \n \n \n m<\/mml:mi>\n m<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n th powers of whose terms are never of the form\n \n \n \n \n \n 2<\/mml:mn>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u00b1\n \n <\/mml:mo>\n \n p<\/mml:mi>\n \n a<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n 2^{n}\\pm p^{a}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with\n \n \n \n \n a<\/mml:mi>\n ,<\/mml:mo>\n n<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n {<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n }<\/mml:mo>\n <\/mml:mrow>\n a,n\\in \\{0,1,2,\\ldots \\}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n p<\/mml:mi>\n p<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n a prime. We also construct another cover of\n \n \n \n \n Z<\/mml:mi>\n <\/mml:mrow>\n \\mathbb {Z}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with odd moduli and use it to prove that\n \n \n \n \n \n x<\/mml:mi>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u2212\n \n <\/mml:mo>\n \n F<\/mml:mi>\n \n 3<\/mml:mn>\n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n x^{2}-F_{3n}\/2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has at least two distinct prime factors whenever\n \n \n \n \n n<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n {<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n }<\/mml:mo>\n <\/mml:mrow>\n n\\in \\{0,1,2,\\ldots \\}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n x<\/mml:mi>\n \n \u2261\n \n <\/mml:mo>\n a<\/mml:mi>\n (<\/mml:mo>\n mod<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n M<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n x\\equiv a (\\operatorname {mod} M)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , where\n \n \n \n \n {<\/mml:mo>\n \n F<\/mml:mi>\n \n i<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n \n }<\/mml:mo>\n \n i<\/mml:mi>\n \n \u2a7e\n \n <\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n \\{F_{i}\\}_{i\\geqslant 0}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the Fibonacci sequence, and\n \n \n \n a<\/mml:mi>\n a<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n M<\/mml:mi>\n M<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n are suitable positive integers having 80 decimal digits.\n <\/p>","DOI":"10.1090\/s0025-5718-09-02212-1","type":"journal-article","created":{"date-parts":[[2009,4,27]],"date-time":"2009-04-27T13:47:33Z","timestamp":1240840053000},"page":"1853-1866","source":"Crossref","is-referenced-by-count":9,"title":["Covers of the integers with odd moduli and their applications to the forms \ud835\udc65^{\ud835\udc5a}-2\u207f and \ud835\udc65\u00b2-\ud835\udc39_{3\ud835\udc5b}\/2"],"prefix":"10.1090","volume":"78","author":[{"given":"Ke-Jian","family":"Wu","sequence":"first","affiliation":[]},{"given":"Zhi-Wei","family":"Sun","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2009,1,30]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"145","DOI":"10.4064\/aa127-2-4","article-title":"On the equation \ud835\udc65\u00b2+\ud835\udc51\ud835\udc66\u00b2=\ud835\udc39_{\ud835\udc5b}","volume":"127","author":"Ballot, Christian","year":"2007","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"2","unstructured":"[B] A. 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