{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:55:09Z","timestamp":1776794109582,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"276","license":[{"start":{"date-parts":[[2012,3,11]],"date-time":"2012-03-11T00:00:00Z","timestamp":1331424000000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Generalized Cullen Numbers are positive integers of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper C Subscript b Baseline left-parenthesis n right-parenthesis colon equals n b Superscript n plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>C<\/mml:mi>\n                              <mml:mi>b<\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo>:=<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">C_b(n):=nb^n+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In this work we generalize some known divisibility properties of Cullen Numbers and present two primality tests for this family of integers. The first test is based in the following property of primes from this family:\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n Superscript b Super Superscript n Baseline identical-to left-parenthesis negative 1 right-parenthesis Superscript b\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>n<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msup>\n                                  <mml:mi>b<\/mml:mi>\n                                  <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                    <mml:mi>n<\/mml:mi>\n                                  <\/mml:mrow>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2261\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>b<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n^{b^{n}}\\equiv (-1)^{b}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    (mod\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n b Superscript n plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:msup>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mi>n<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">nb^n+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ). It is stronger and has less computational cost than Fermat\u2019s test (to bases\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b\">\n                        <mml:semantics>\n                          <mml:mi>b<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">b<\/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=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ) and than Miller-Rabin\u2019s test (if\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b\">\n                        <mml:semantics>\n                          <mml:mi>b<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">b<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is odd, to base\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n\">\n                        <mml:semantics>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">n<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ). Pseudoprimes for this new test seem to be very scarce, only 4 pseudoprimes have been found among the many millions of Generalized Cullen Numbers tested. We also present a second, more demanding, test for which no pseudoprimes have been found. These tests lead to an algorithm, running in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"ModifyingAbove upper O With tilde left-parenthesis log squared left-parenthesis upper N right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mover>\n                                <mml:mi>O<\/mml:mi>\n                                <mml:mo stretchy=\"false\">\n                                  ~\n                                  \n                                <\/mml:mo>\n                              <\/mml:mover>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>log<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\tilde {O}(\\log ^2(N))<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    time, which might be very useful in the search of Generalized Cullen Primes.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2011-02489-0","type":"journal-article","created":{"date-parts":[[2011,3,11]],"date-time":"2011-03-11T08:51:42Z","timestamp":1299833502000},"page":"2315-2323","source":"Crossref","is-referenced-by-count":4,"title":["An \ud835\udc42\u0303(log\u00b2(\ud835\udc41)) time primality test for generalized Cullen numbers"],"prefix":"10.1090","volume":"80","author":[{"given":"Jos\u00e9","family":"Grau","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Antonio","family":"Oller-Marc\u00e9n","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2011,3,11]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"173","DOI":"10.2307\/2006975","article-title":"On distinguishing prime numbers from composite numbers","volume":"117","author":"Adleman, Leonard M.","year":"1983","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"781","DOI":"10.4007\/annals.2004.160.781","article-title":"PRIMES is in P","volume":"160","author":"Agrawal, Manindra","year":"2004","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"issue":"252","key":"3","doi-asserted-by":"publisher","first-page":"2043","DOI":"10.1090\/S0025-5718-05-01727-8","article-title":"Sharpening \u201cPRIMES is in \ud835\udc43\u201d for a large family of numbers","volume":"74","author":"Berrizbeitia, Pedro","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","unstructured":"Pedro Berrizbeitia and Jos\u00e9 Gregorio Fernandes, Observaciones sobre la primalidad de los n\u00fameros de cullen, Short communication in \u201cTerceras Jornadas de Teor\u00eda de N\u00fameros\u201d (http:\/\/campus.usal.es\/ tjtn2009\/doc\/abstracts.pdf)."},{"issue":"165","key":"5","doi-asserted-by":"publisher","first-page":"297","DOI":"10.2307\/2007581","article-title":"Primality testing and Jacobi sums","volume":"42","author":"Cohen, H.","year":"1984","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"6","unstructured":"James Cullen, Question 15897, Educ. Times (1905), no. Dec., 534."},{"key":"7","unstructured":"A. Cunningham and H.J. Woodall, Factorisation of \ud835\udc44=(2^{\ud835\udc5e}\u2213\ud835\udc5e) and (\ud835\udc5e2^{\ud835\udc5e}\u22131), Messenger Math. 47 (1917), 1\u201338."},{"key":"8","unstructured":"Harvey Dubner, Generalized Cullen numbers, J. Recreat. Math. 21 (1989), 190\u2013194."},{"key":"9","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":"10","series-title":"Cambridge Tracts in Mathematics, No. 70","volume-title":"Applications of sieve methods to the theory of numbers","author":"Hooley, C.","year":"1976"},{"issue":"212","key":"11","doi-asserted-by":"publisher","first-page":"1733","DOI":"10.2307\/2153382","article-title":"New Cullen primes","volume":"64","author":"Keller, Wilfrid","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"12","isbn-type":"print","doi-asserted-by":"publisher","first-page":"176","DOI":"10.1007\/978-3-642-60408-9_15","article-title":"On primes recognizable in deterministic polynomial time","author":"Konyagin, Sergei","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/3540610324"},{"key":"13","series-title":"CMS Books in Mathematics\/Ouvrages de Math\\'{e}matiques de la SMC","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-21850-2","volume-title":"17 lectures on Fermat numbers","volume":"9","author":"K\u0159\u00ed\u017eek, Michal","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0387953329"},{"key":"14","doi-asserted-by":"publisher","first-page":"253","DOI":"10.1007\/BF02941281","article-title":"On the greatest common divisor of two Cullen numbers","volume":"73","author":"Luca, F.","year":"2003","journal-title":"Abh. Math. Sem. Univ. 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Rechnen)","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"key":"20","series-title":"Canadian Mathematical Society Series of Monographs and Advanced Texts","isbn-type":"print","volume-title":"\\'{E}douard Lucas and primality testing","volume":"22","author":"Williams, Hugh C.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0471148520"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02489-0\/S0025-5718-2011-02489-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02489-0\/S0025-5718-2011-02489-0.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:58:11Z","timestamp":1776790691000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2011-80-276\/S0025-5718-2011-02489-0\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,3,11]]},"references-count":20,"journal-issue":{"issue":"276","published-print":{"date-parts":[[2011,10]]}},"alternative-id":["S0025-5718-2011-02489-0"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2011-02489-0","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":[[2011,3,11]]}}}