{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T16:25:12Z","timestamp":1781108712110,"version":"3.54.1"},"reference-count":32,"publisher":"American Mathematical Society (AMS)","issue":"231","license":[{"start":{"date-parts":[[2001,3,1]],"date-time":"2001-03-01T00:00:00Z","timestamp":983404800000},"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":"<p>\n                    We report the discovery of a new factor for each of the Fermat numbers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F 13 comma upper F 15 comma upper F 16\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>13<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>15<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>16<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">F_{13}, F_{15}, F_{16}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . These new factors have 27, 33 and 27 decimal digits respectively. Each factor was found by the elliptic curve method. After division by the new factors and previously known factors, the remaining cofactors are seen to be composite numbers with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2391\">\n                        <mml:semantics>\n                          <mml:mn>2391<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">2391<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"9808\">\n                        <mml:semantics>\n                          <mml:mn>9808<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">9808<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"19694\">\n                        <mml:semantics>\n                          <mml:mn>19694<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">19694<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    decimal digits respectively.\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01207-2","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"1297-1304","source":"Crossref","is-referenced-by-count":4,"title":["Three new factors of Fermat numbers"],"prefix":"10.1090","volume":"69","author":[{"given":"R.","family":"Brent","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R.","family":"Crandall","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"K.","family":"Dilcher","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C.","family":"Van Halewyn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2000,3,1]]},"reference":[{"key":"1","unstructured":"R. P. Brent, Some integer factorization algorithms using elliptic curves, Australian Computer Science Communications 8 (1986), 149\u2013163. Also Report CMA-R32-85, Centre for Mathematical Analysis, Australian National University, Canberra, Sept. 1985, 20 pp."},{"key":"2","unstructured":"R. P. Brent, Factorization of the eleventh Fermat number (preliminary report), AMS Abstracts 10 (1989), 89T-11-73."},{"key":"3","unstructured":"R. P. Brent, Factorization of the tenth and eleventh Fermat numbers, Report TR-CS-96-02, Computer Sciences Laboratory, Australian National Univ., Canberra, Feb. 1997. \\url{ftp:\/\/nimbus.anu.edu.au\/pub\/Brent\/rpb161tr.dvi.gz}."},{"issue":"225","key":"4","doi-asserted-by":"publisher","first-page":"429","DOI":"10.1090\/S0025-5718-99-00992-8","article-title":"Factorization of the tenth Fermat number","volume":"68","author":"Brent, Richard P.","year":"1999","journal-title":"Math. 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Suyama, Informal preliminary report (8), personal communication to Brent, October 1985."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2000-69-231\/S0025-5718-00-01207-2\/S0025-5718-00-01207-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-231\/S0025-5718-00-01207-2\/S0025-5718-00-01207-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:22:44Z","timestamp":1776723764000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-231\/S0025-5718-00-01207-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,3,1]]},"references-count":32,"journal-issue":{"issue":"231","published-print":{"date-parts":[[2000,7]]}},"alternative-id":["S0025-5718-00-01207-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01207-2","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2000,3,1]]}}}