{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:47:08Z","timestamp":1776844028859,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"241","license":[{"start":{"date-parts":[[2003,5,2]],"date-time":"2003-05-02T00:00:00Z","timestamp":1051833600000},"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 Hardy and Littlewood\u2019s Conjecture F implies that the asymptotic density of prime values of the polynomials\n \n \n \n \n \n f<\/mml:mi>\n A<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n \n x<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n +<\/mml:mo>\n x<\/mml:mi>\n +<\/mml:mo>\n A<\/mml:mi>\n ,<\/mml:mo>\n <\/mml:mrow>\n f_A(x) = x^2 + x + A,<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n \n \n \n \n A<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n \n Z<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n A \\in \\mathbb {Z}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , is related to the discriminant\n \n \n \n \n \n \u0394\n \n <\/mml:mi>\n =<\/mml:mo>\n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n 4<\/mml:mn>\n A<\/mml:mi>\n <\/mml:mrow>\n \\Delta = 1 - 4A<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n \n \n f<\/mml:mi>\n A<\/mml:mi>\n <\/mml:msub>\n (<\/mml:mo>\n x<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n f_A(x)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n via a quantity\n \n \n \n \n C<\/mml:mi>\n (<\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n )<\/mml:mo>\n .<\/mml:mo>\n <\/mml:mrow>\n C(\\Delta ).<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n The larger\n \n \n \n \n C<\/mml:mi>\n (<\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n C(\\Delta )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is, the higher the asymptotic density of prime values for any quadratic polynomial of discriminant\n \n \n \n \n \u0394\n \n <\/mml:mi>\n \\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . A technique of Bach allows one to estimate\n \n \n \n \n C<\/mml:mi>\n (<\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n C(\\Delta )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n accurately for any\n \n \n \n \n \n \u0394\n \n <\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\Delta > 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , given the class number of the imaginary quadratic order with discriminant\n \n \n \n \n \u0394\n \n <\/mml:mi>\n \\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and for any\n \n \n \n \n \n \u0394\n \n <\/mml:mi>\n ><\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n \\Delta > 0<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n given the class number and regulator of the real quadratic order with discriminant\n \n \n \n \n \u0394\n \n <\/mml:mi>\n \\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The Manitoba Scalable Sieve Unit (MSSU) has shown us how to rapidly generate many discriminants\n \n \n \n \n \u0394\n \n <\/mml:mi>\n \\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for which\n \n \n \n \n C<\/mml:mi>\n (<\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n C(\\Delta )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is potentially large, and new methods for evaluating class numbers and regulators of quadratic orders allow us to compute accurate estimates of\n \n \n \n \n C<\/mml:mi>\n (<\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n C(\\Delta )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n efficiently, even for values of\n \n \n \n \n \u0394\n \n <\/mml:mi>\n \\Delta<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with as many as\n \n \n \n 70<\/mml:mn>\n 70<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n decimal digits. Using these methods, we were able to find a number of discriminants for which, under the assumption of the Extended Riemann Hypothesis,\n \n \n \n \n C<\/mml:mi>\n (<\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n C(\\Delta )<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is larger than any previously known examples.\n <\/p>","DOI":"10.1090\/s0025-5718-02-01418-7","type":"journal-article","created":{"date-parts":[[2002,11,1]],"date-time":"2002-11-01T09:59:50Z","timestamp":1036144790000},"page":"499-519","source":"Crossref","is-referenced-by-count":16,"title":["New quadratic polynomials with high densities of prime values"],"prefix":"10.1090","volume":"72","author":[{"suffix":"Jr.","given":"Michael","family":"Jacobson","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hugh","family":"Williams","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2002,5,2]]},"reference":[{"key":"1","unstructured":"C.S. Abel, Ein Algorithmus zur Berechnung der Klassenzahl und des Regulators reellquadratischer Ordnungen, Ph.D. thesis, Universit\u00e4t des Saarlandes, Saarbr\u00fccken, Germany, 1994."},{"key":"2","isbn-type":"print","first-page":"13","article-title":"Improved approximations for Euler products","author":"Bach, Eric","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0821803123"},{"key":"3","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1090\/gsm\/058","article-title":"The 7-15 problem","volume":"9","author":"Venkatarayudu, T.","year":"1939","journal-title":"Proc. Indian Acad. Sci., Sect. A."},{"key":"4","isbn-type":"print","first-page":"27","article-title":"A subexponential algorithm for the determination of class groups and regulators of algebraic number fields","author":"Buchmann, Johannes","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0817634932"},{"key":"5","isbn-type":"print","first-page":"35","article-title":"Calculs de nombres de classes et de r\u00e9gulateurs de corps quadratiques en temps sous-exponentiel","author":"Cohen, H.","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/0817636846"},{"key":"6","unstructured":"S. D\u00fcllmann, Ein Algorithmus zur Bestimmung der Klassengruppe positiv definiter bin\u00e4rer quadratischer Formen, Ph.D. thesis, Universit\u00e4t des Saarlandes, Saarbr\u00fccken, Germany, 1991."},{"issue":"191","key":"7","doi-asserted-by":"publisher","first-page":"345","DOI":"10.2307\/2008810","article-title":"Quadratic polynomials which have a high density of prime values","volume":"55","author":"Fung, G. W.","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","unstructured":"The LiDIA Group, LiDIA: a c++ library for computational number theory, Software, Technische Universit\u00e4t Darmstadt, Germany, 1997, See http:\/\/www.informatik.tu-darmstadt.de\/TI\/LiDIA."},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"837","DOI":"10.2307\/1990896","article-title":"A rigorous subexponential algorithm for computation of class groups","volume":"2","author":"Hafner, James L.","year":"1989","journal-title":"J. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0894-0347","issn-type":"print"},{"key":"10","doi-asserted-by":"crossref","unstructured":"G.H. Hardy and J.E. Littlewood, Partitio numerorum III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1\u201370.","DOI":"10.1007\/BF02403921"},{"key":"11","unstructured":"M.J. Jacobson, Jr., Computational techniques in quadratic fields, Master\u2019s thesis, University of Manitoba, Winnipeg, Manitoba, 1995."},{"key":"12","unstructured":"\\bysame, Subexponential class group computation in quadratic orders, Ph.D. thesis, Technische Universit\u00e4t Darmstadt, Darmstadt, Germany, 1999."},{"key":"13","doi-asserted-by":"publisher","first-page":"58","DOI":"10.1016\/0022-314X(70)90006-5","article-title":"The size of \ud835\udc3f(1,\ud835\udf12) for real nonprincipal residue characters \ud835\udf12 with prime modulus","volume":"2","author":"Joshi, Padmini Tryambak","year":"1970","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"14","unstructured":"D.H. Lehmer, On the function \ud835\udc65\u00b2+\ud835\udc65+\ud835\udc34, Sphinx 6 (1937), 212\u2013214, 1936 and 7:40."},{"key":"15","doi-asserted-by":"publisher","first-page":"433","DOI":"10.2307\/2004491","article-title":"Integer sequences having prescribed quadratic character","volume":"24","author":"Lehmer, D. H.","year":"1970","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"16","doi-asserted-by":"crossref","unstructured":"J.E. Littlewood, On the class number of the corpus \ud835\udc43(\u221a-\ud835\udc58), Proc. London Math. Soc. 27 (1928), 358\u2013372.","DOI":"10.1112\/plms\/s2-27.1.358"},{"key":"17","unstructured":"R.F. Lukes, A very fast electronic number sieve, Ph.D. thesis, University of Manitoba, Winnipeg, Manitoba, 1995."},{"issue":"1","key":"18","first-page":"113","article-title":"Numerical sieving devices: their history and some applications","volume":"13","author":"Lukes, R. 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