{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T15:23:12Z","timestamp":1773242592210,"version":"3.50.1"},"reference-count":21,"publisher":"Wiley","issue":"1","license":[{"start":{"date-parts":[[2015,1,1]],"date-time":"2015-01-01T00:00:00Z","timestamp":1420070400000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2015]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>The discriminant of a trinomial of the form <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline1\"\/><jats:tex-math>$x^{n}\\pm \\,x^{m}\\pm \\,1$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> has the form <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline2\"\/><jats:tex-math>$\\pm n^{n}\\pm (n-m)^{n-m}m^{m}$<\/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\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline3\"\/><jats:tex-math>$n$<\/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\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline4\"\/><jats:tex-math>$m$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> are relatively prime. We investigate when these discriminants have nontrivial square factors. We explain various unlikely-seeming parametric families of square factors of these discriminant values: for example, when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline5\"\/><jats:tex-math>$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is congruent to 2 (mod 6) we have that <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline6\"\/><jats:tex-math>$((n^{2}-n+1)\/3)^{2}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> always divides <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline7\"\/><jats:tex-math>$n^{n}-(n-1)^{n-1}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>. In addition, we discover many other square factors of these discriminants that do not fit into these parametric families. The set of primes whose squares can divide these sporadic values as <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline8\"\/><jats:tex-math>$n$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> varies seems to be independent of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline9\"\/><jats:tex-math>$m$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>, and this set can be seen as a generalization of the Wieferich primes, those primes <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline10\"\/><jats:tex-math>$p$<\/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\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline11\"\/><jats:tex-math>$2^{p}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula> is congruent to 2 (mod <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" mime-subtype=\"gif\" xlink:type=\"simple\" xlink:href=\"S1461157014000436_inline12\"\/><jats:tex-math>$p^{2}$<\/jats:tex-math><\/jats:alternatives><\/jats:inline-formula>). We provide heuristics for the density of these sporadic primes and the density of squarefree values of these trinomial discriminants.<\/jats:p>","DOI":"10.1112\/s1461157014000436","type":"journal-article","created":{"date-parts":[[2015,1,27]],"date-time":"2015-01-27T04:15:56Z","timestamp":1422332156000},"page":"148-169","source":"Crossref","is-referenced-by-count":15,"title":["Squarefree values of trinomial discriminants"],"prefix":"10.1112","volume":"18","author":[{"given":"David W.","family":"Boyd","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Greg","family":"Martin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mark","family":"Thom","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2015,1,1]]},"reference":[{"key":"S1461157014000436_r5","first-page":"451","volume-title":"Analytic number theory: Proceedings of a conference in honor of Heini Halberstam","author":"Heath\u2013Brown","year":"1996"},{"key":"S1461157014000436_r12","first-page":"52","article-title":"An estimate for the number of roots of some comparisons by the Stepanov method","volume":"51","author":"Mit\u2019kin","year":"1992","journal-title":"Mat. 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