{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,13]],"date-time":"2026-07-13T23:22:23Z","timestamp":1783984943697,"version":"3.55.0"},"reference-count":21,"publisher":"American Mathematical Society (AMS)","issue":"302","license":[{"start":{"date-parts":[[2017,3,24]],"date-time":"2017-03-24T00:00:00Z","timestamp":1490313600000},"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                    Kurepa\u2019s conjecture states that there is no odd prime\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    that divides\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"factorial p equals 0 factorial plus 1 factorial plus midline-horizontal-ellipsis plus left-parenthesis p minus 1 right-parenthesis factorial\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo>!<\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>!<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>!<\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo>\n                              \u22ef\n                              \n                            <\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>!<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">!p=0!+1!+\\cdots +(p-1)!<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We search for a counterexample to this conjecture for all\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p greater-than 2 Superscript 34\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>34<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p&gt;2^{34}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We introduce new optimization techniques and perform the computation using graphics processing units. Additionally, we consider the generalized Kurepa\u2019s left factorial given by\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"factorial Superscript k Baseline n equals left-parenthesis 0 factorial right-parenthesis Superscript k Baseline plus left-parenthesis 1 factorial right-parenthesis Superscript k Baseline plus midline-horizontal-ellipsis plus left-parenthesis left-parenthesis n minus 1 right-parenthesis factorial right-parenthesis Superscript k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mo>!<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>!<\/mml:mo>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>!<\/mml:mo>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo>\n                              \u22ef\n                              \n                            <\/mml:mo>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>!<\/mml:mo>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">!^{k}n=(0!)^k +(1!)^k +\\cdots +((n-1)!)^{k}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and show that for all integers\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"1 greater-than k greater-than 100\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>100<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">1&gt;k&gt;100<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    there exists an odd prime\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p\">\n                        <mml:semantics>\n                          <mml:mi>p<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p bar factorial Superscript k Baseline p\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>\n                              \u2223\n                              \n                            <\/mml:mo>\n                            <mml:msup>\n                              <mml:mo>!<\/mml:mo>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mi>p<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p\\mid !^k p<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3098","type":"journal-article","created":{"date-parts":[[2015,8,28]],"date-time":"2015-08-28T07:45:13Z","timestamp":1440747913000},"page":"3061-3068","source":"Crossref","is-referenced-by-count":5,"title":["Searching for a counterexample to Kurepa\u2019s conjecture"],"prefix":"10.1090","volume":"85","author":[{"given":"Vladica","family":"Andreji\u0107","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Milos","family":"Tatarevic","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2016,3,24]]},"reference":[{"key":"1","unstructured":"AMD Inc., AMD Accelerated Parallel Processing OpenCL Programming Guide, revision 2.7, (2013)"},{"key":"2","doi-asserted-by":"crossref","unstructured":"H. G. Backer, Computing A*B (mod N) Efficiently in ANSI C, ACM Sigplan Notices 27 (1992), 95\u201398.","DOI":"10.1145\/130722.130735"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.5802\/jtnb.432","article-title":"Nombres de Bell et somme de factorielles","volume":"16","author":"Barsky, Daniel","year":"2004","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"527","DOI":"10.5802\/jtnb.775","article-title":"Erratum \u00e0 l\u2019article Nombres de Bell et somme de factorielles [MR2145571]","volume":"23","author":"Barsky, Daniel","year":"2011","journal-title":"J. Th\\'{e}or. Nombres Bordeaux","ISSN":"https:\/\/id.crossref.org\/issn\/1246-7405","issn-type":"print"},{"key":"5","unstructured":"K. Brown, Can \ud835\udc5b Divide !\ud835\udc5b ?, \\url{http:\/\/www.mathpages.com\/home\/kmath064.htm}."},{"issue":"290","key":"6","doi-asserted-by":"publisher","first-page":"3071","DOI":"10.1090\/S0025-5718-2014-02800-7","article-title":"A search for Wilson primes","volume":"83","author":"Costa, Edgar","year":"2014","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"217","key":"7","doi-asserted-by":"publisher","first-page":"433","DOI":"10.1090\/S0025-5718-97-00791-6","article-title":"A search for Wieferich and Wilson primes","volume":"66","author":"Crandall, Richard","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"8","unstructured":"Y. Gallot, Is the number of primes \\frac{1}2\u2211\u1d62\u208c\u2080\u207f\u207b\u00b9\ud835\udc56! finite, http:\/\/yves.gallot.pagesperso- orange.fr, (2000)."},{"key":"9","unstructured":"G. Gogi\u0107, Parallel Algorithms in Arithmetic, Master thesis, Belgrade University, (1991)"},{"key":"10","series-title":"Problem Books in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-26677-0","volume-title":"Unsolved problems in number theory","author":"Guy, Richard K.","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0387208607","edition":"3"},{"key":"11","doi-asserted-by":"crossref","unstructured":"I. S. Haque and V. S. Pande, Hard Data on Soft Errors: A Large-Scale Assessment of Real-World Error Rates in GPGPU, Proceedings of the 2010 10th IEEE\/ACM International Conference on Cluster, Cloud and Grid Computing (2010), 691\u2013696.","DOI":"10.1109\/CCGRID.2010.84"},{"key":"12","first-page":"19","article-title":"On Kurepa\u2019s problems in number theory","volume":"57(71)","author":"Ivi\u0107, A.","year":"1995","journal-title":"Publ. Inst. Math. (Beograd) (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0350-1302","issn-type":"print"},{"key":"13","unstructured":"P. Jobling, A couple of searches, \\url{https:\/\/groups.yahoo.com\/neo\/groups\/primeform\/conversations\/topics\/5095}, (2004)."},{"key":"14","first-page":"147","article-title":"On the left factorial function !\ud835\udc5b","volume":"1","author":"Kurepa, \u0110uro","year":"1971","journal-title":"Math. Balkanica","ISSN":"https:\/\/id.crossref.org\/issn\/0350-2007","issn-type":"print"},{"key":"15","unstructured":"B. Males\u0306evi\u0107, Private communication."},{"key":"16","first-page":"24","article-title":"On some formulas involving !\ud835\udc5b and the verification of the !\ud835\udc5b-hypothesis by use of computers","volume":"47(61)","author":"Mijajlovi\u0107, \u017d.","year":"1990","journal-title":"Publ. Inst. Math. (Beograd) (N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0350-1302","issn-type":"print"},{"issue":"170","key":"17","doi-asserted-by":"publisher","first-page":"519","DOI":"10.2307\/2007970","article-title":"Modular multiplication without trial division","volume":"44","author":"Montgomery, Peter L.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"18","unstructured":"OEIS Foundation Inc. (2011), The On-Line Encyclopedia of Integer Sequences, \\url{http:\/\/oeis.org\/A000166}"},{"key":"19","unstructured":"PrimeGrid, Wall-Sun-Sun Prime Search, \\url{http:\/\/www.primegrid.com}, March 2014."},{"key":"20","unstructured":"PrimeGrid, Wieferich Prime Search, \\url{http:\/\/www.primegrid.com}, August 2014."},{"issue":"225","key":"21","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1090\/S0025-5718-99-00990-4","article-title":"The number of primes \u2211\u207f\u1d62\u208c\u2081(-1)\u207f\u207b\u2071\ud835\udc56! is finite","volume":"68","author":"\u017divkovi\u0107, Miodrag","year":"1999","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\/2016-85-302\/S0025-5718-2016-03098-7\/S0025-5718-2016-03098-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-302\/S0025-5718-2016-03098-7\/S0025-5718-2016-03098-7.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:57:14Z","timestamp":1776797834000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2016-85-302\/S0025-5718-2016-03098-7\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,3,24]]},"references-count":21,"journal-issue":{"issue":"302","published-print":{"date-parts":[[2016,11]]}},"alternative-id":["S0025-5718-2016-03098-7"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3098","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":[[2016,3,24]]}}}