{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,8]],"date-time":"2026-09-08T08:25:34Z","timestamp":1788855934704,"version":"build-2803163510"},"reference-count":16,"publisher":"National Academy of Sciences","issue":"31","content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2016,8,2]]},"abstract":"<jats:title>Significance<\/jats:title>\n                  <jats:p>\n                    Prime numbers play a central role in analytic number theory, and are well known to be very well distributed among the reduced residue classes\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" overflow=\"scroll\">\n                        <mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:mrow>\n                              <mml:mtext>mod<\/mml:mtext>\n                              <mml:mo>\u2009<\/mml:mo>\n                              <mml:mi>q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    . Surprisingly, the same does not appear to be true for sequences of consecutive primes, with different patterns occurring with wildly different frequencies. We formulate a precise conjecture, based on the Hardy\u2212Littlewood conjectures, which explains this phenomenon. In particular, we predict that all patterns do occur their fair share of the time in the limit, but that there are secondary terms only very slowly tending to zero that create the observed biases.\n                  <\/jats:p>","DOI":"10.1073\/pnas.1605366113","type":"journal-article","created":{"date-parts":[[2016,7,14]],"date-time":"2016-07-14T23:14:28Z","timestamp":1468538068000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":38,"title":["Unexpected biases in the distribution of consecutive primes"],"prefix":"10.1073","volume":"113","author":[{"given":"Robert J.","family":"Lemke Oliver","sequence":"first","affiliation":[{"name":"Department of Mathematics, Stanford University, Stanford, CA 94305;"},{"name":"Department of Mathematics, Tufts University, Medford, MA 02155"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kannan","family":"Soundararajan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Stanford University, Stanford, CA 94305;"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2016,7,14]]},"reference":[{"key":"e_1_3_3_1_2","doi-asserted-by":"publisher","DOI":"10.1080\/10586458.1994.10504289"},{"key":"e_1_3_3_2_2","doi-asserted-by":"publisher","DOI":"10.1080\/00029890.2006.11920275"},{"key":"e_1_3_3_3_2","first-page":"157","volume-title":"Number Theory and Algebra","author":"Knapowski S","year":"1977","unstructured":"S Knapowski, P Tur\u00e1n, On prime numbers \u22611 resp. 3mod 4. 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