{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:09:04Z","timestamp":1776845344015,"version":"3.51.2"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"290","license":[{"start":{"date-parts":[[2015,5,5]],"date-time":"2015-05-05T00:00:00Z","timestamp":1430784000000},"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                    Define\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"psi Subscript m\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03c8\n                              \n                            <\/mml:mi>\n                            <mml:mi>m<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\psi _m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to be the smallest strong pseudoprime to the first\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"m\">\n                        <mml:semantics>\n                          <mml:mi>m<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    prime bases. The exact value of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"psi Subscript m\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03c8\n                              \n                            <\/mml:mi>\n                            <mml:mi>m<\/mml:mi>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\psi _m<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is known for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1 less-than-or-equal-to m less-than-or-equal-to 8\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:mn>8<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">1\\le m \\le 8<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Z. Zhang has found a 19-decimal-digit number\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper Q 11 equals 3825 12305 65464 13051\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>Q<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>11<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>3825<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>12305<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>65464<\/mml:mn>\n                            <mml:mspace width=\"thinmathspace\"\/>\n                            <mml:mn>13051<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">Q_{11}=3825\\,12305\\,65464\\,13051<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    which is a strong pseudoprime to the first 11 prime bases and he conjectured that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"psi 9 equals psi 10 equals psi 11 equals upper Q 11 period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03c8\n                                \n                              <\/mml:mi>\n                              <mml:mn>9<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03c8\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>10<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\n                                \u03c8\n                                \n                              <\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>11<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>Q<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>11<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>.<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\psi _9=\\psi _{10}=\\psi _{11}=Q_{11}.<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    We tabulate all the strong pseudoprimes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n less-than-or-equal-to upper Q 11\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2264\n                              \n                            <\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>Q<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>11<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\le Q_{11}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    to the first eight prime bases, and prove Zhang\u2019s conjecture.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2014-02830-5","type":"journal-article","created":{"date-parts":[[2014,5,5]],"date-time":"2014-05-05T10:18:47Z","timestamp":1399285127000},"page":"2915-2924","source":"Crossref","is-referenced-by-count":6,"title":["Strong pseudoprimes to the first eight prime bases"],"prefix":"10.1090","volume":"83","author":[{"given":"Yupeng","family":"Jiang","sequence":"first","affiliation":[]},{"given":"Yingpu","family":"Deng","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2014,5,5]]},"reference":[{"issue":"9","key":"1","first-page":"Article 11.9.2, 14","article-title":"A Wieferich prime search up to 6.7\u00d710\u00b9\u2075","volume":"14","author":"Dorais, Fran\u00e7ois G.","year":"2011","journal-title":"J. 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Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"250","key":"7","doi-asserted-by":"publisher","first-page":"1009","DOI":"10.1090\/S0025-5718-04-01693-X","article-title":"Finding \ud835\udc36\u2083-strong pseudoprimes","volume":"74","author":"Zhang, Zhenxiang","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"260","key":"8","doi-asserted-by":"publisher","first-page":"2095","DOI":"10.1090\/S0025-5718-07-01977-1","article-title":"Two kinds of strong pseudoprimes up to 10\u00b3\u2076","volume":"76","author":"Zhang, Zhenxiang","year":"2007","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"244","key":"9","doi-asserted-by":"publisher","first-page":"2085","DOI":"10.1090\/S0025-5718-03-01545-X","article-title":"Finding strong pseudoprimes to several bases. II","volume":"72","author":"Zhang, Zhenxiang","year":"2003","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2014-83-290\/S0025-5718-2014-02830-5\/S0025-5718-2014-02830-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-290\/S0025-5718-2014-02830-5\/S0025-5718-2014-02830-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:07:53Z","timestamp":1776794873000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-290\/S0025-5718-2014-02830-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,5,5]]},"references-count":9,"journal-issue":{"issue":"290","published-print":{"date-parts":[[2014,11]]}},"alternative-id":["S0025-5718-2014-02830-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2014-02830-5","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":[[2014,5,5]]}}}