{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T12:38:51Z","timestamp":1773232731770,"version":"3.50.1"},"reference-count":29,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2014,4,1]],"date-time":"2014-04-01T00:00:00Z","timestamp":1396310400000},"content-version":"unspecified","delay-in-days":90,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline1\"\/><jats:tex-math>$Q(N;q,a)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> be the number of squares in the arithmetic progression <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline2\"\/><jats:tex-math>$qn+a$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline3\"\/><jats:tex-math>$n=0$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>,<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline4\"\/><jats:tex-math>$1,\\ldots,N-1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and let <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline5\"\/><jats:tex-math>$Q(N)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> be the maximum of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline6\"\/><jats:tex-math>$Q(N;q,a)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> over all non-trivial arithmetic progressions <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline7\"\/><jats:tex-math>$qn + a$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. Rudin\u2019s conjecture claims that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline8\"\/><jats:tex-math>$Q(N)=O(\\sqrt{N})$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and in its stronger form that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline9\"\/><jats:tex-math>$Q(N)=Q(N;24,1)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> if <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline10\"\/><jats:tex-math>$N\\ge 6$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We prove the conjecture above for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline11\"\/><jats:tex-math>$6\\le N\\le 52$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. We even prove that the arithmetic progression <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline12\"\/><jats:tex-math>$24n+1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is the only one, up to equivalence, that contains <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline13\"\/><jats:tex-math>$Q(N)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> squares for the values of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline14\"\/><jats:tex-math>$N$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> such that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline15\"\/><jats:tex-math>$Q(N)$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> increases, for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline16\"\/><jats:tex-math>$7\\le N\\le 52$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline17\"\/><jats:tex-math>$N=8,13,16,23,27,36,41$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>  and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" mimetype=\"image\" xlink:type=\"simple\" xlink:href=\"S1461157013000259_inline18\"\/><jats:tex-math>$52$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>).<\/jats:p><jats:p><jats:uri xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:type=\"simple\" xlink:href=\"http:\/\/journals.cambridge.org\/sup_S1461157013000259sup001\">Supplementary\u00a0materials\u00a0are\u00a0available\u00a0with\u00a0this\u00a0article.<\/jats:uri><\/jats:p>","DOI":"10.1112\/s1461157013000259","type":"journal-article","created":{"date-parts":[[2014,4,25]],"date-time":"2014-04-25T04:49:42Z","timestamp":1398401382000},"page":"58-76","source":"Crossref","is-referenced-by-count":4,"title":["On a conjecture of Rudin on squares in arithmetic progressions"],"prefix":"10.1112","volume":"17","author":[{"given":"Enrique","family":"Gonz\u00e1lez-Jim\u00e9nez","sequence":"first","affiliation":[]},{"given":"Xavier","family":"Xarles","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2014,4,1]]},"reference":[{"key":"S1461157013000259_r15","doi-asserted-by":"publisher","DOI":"10.1023\/A:1000111601294"},{"key":"S1461157013000259_r28","unstructured":"28. 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