{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:53:55Z","timestamp":1776848035225,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"238","license":[{"start":{"date-parts":[[2002,5,17]],"date-time":"2002-05-17T00:00:00Z","timestamp":1021593600000},"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                    Numbers of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper F Subscript b comma n Baseline equals b Superscript 2 Super Superscript n Baseline plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>F<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mi>b<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>n<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>b<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:msup>\n                                  <mml:mn>2<\/mml:mn>\n                                  <mml:mi>n<\/mml:mi>\n                                <\/mml:msup>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">F_{b,n}=b^{2^n}+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 Superscript m Baseline minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>m<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">2^m-1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . The theoretical distributions of GFN primes, for fixed\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                    , are derived and compared to the actual distributions. The predictions are surprisingly accurate and can be used to support Bateman and Horn\u2019s quantitative form of \u201cHypothesis H\" of Schinzel and Sierpi\u0144ski. A list of the current largest known GFN primes is included.\n                  <\/p>","DOI":"10.1090\/s0025-5718-01-01350-3","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:28Z","timestamp":1027707268000},"page":"825-832","source":"Crossref","is-referenced-by-count":6,"title":["Distribution of generalized Fermat prime numbers"],"prefix":"10.1090","volume":"71","author":[{"given":"Harvey","family":"Dubner","sequence":"first","affiliation":[]},{"given":"Yves","family":"Gallot","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2001,5,17]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"363","DOI":"10.2307\/2004056","article-title":"A heuristic asymptotic formula concerning the distribution of prime numbers","volume":"16","author":"Bateman, Paul T.","year":"1962","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4324-3","volume-title":"Projects in scientific computation","author":"Crandall, Richard E.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387978089"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-2334-4","volume-title":"Topics in advanced scientific computation","author":"Crandall, Richard E.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0387944737"},{"issue":"205","key":"4","doi-asserted-by":"publisher","first-page":"305","DOI":"10.2307\/2153411","article-title":"Discrete weighted transforms and large-integer arithmetic","volume":"62","author":"Crandall, Richard","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"5","volume-title":"Introduction to statistical analysis. 2nd ed","author":"Dixon, Wilfrid J.","year":"1957"},{"issue":"209","key":"6","doi-asserted-by":"publisher","first-page":"397","DOI":"10.2307\/2153343","article-title":"Factors of generalized Fermat numbers","volume":"64","author":"Dubner, Harvey","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"7","isbn-type":"print","volume-title":"An introduction to the theory of numbers","author":"Hardy, G. H.","year":"1979","ISBN":"https:\/\/id.crossref.org\/isbn\/0198531702","edition":"5"},{"key":"8","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0759-7","volume-title":"The new book of prime number records","author":"Ribenboim, Paulo","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0387944575"},{"key":"9","first-page":"320","article-title":"On the conjecture of Hardy & Littlewood concerning the number of primes of the form \ud835\udc5b\u00b2+\ud835\udc4e","volume":"14","author":"Shanks, Daniel","year":"1960","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","first-page":"186","article-title":"On numbers of the form \ud835\udc5b\u2074+1","volume":"15","author":"Shanks, Daniel","year":"1961","journal-title":"Math. Comput."},{"key":"11","doi-asserted-by":"publisher","first-page":"185","DOI":"10.4064\/aa-4-3-185-208","article-title":"Sur certaines hypoth\u00e8ses concernant les nombres premiers","volume":"4","author":"Schinzel, A.","year":"1958","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01350-3\/S0025-5718-01-01350-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01350-3\/S0025-5718-01-01350-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:56:54Z","timestamp":1776725814000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-238\/S0025-5718-01-01350-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,5,17]]},"references-count":11,"journal-issue":{"issue":"238","published-print":{"date-parts":[[2002,4]]}},"alternative-id":["S0025-5718-01-01350-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01350-3","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":[[2001,5,17]]}}}