{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:22:47Z","timestamp":1776784967589,"version":"3.51.2"},"reference-count":17,"publisher":"American Mathematical Society (AMS)","issue":"253","license":[{"start":{"date-parts":[[2006,6,16]],"date-time":"2006-06-16T00:00:00Z","timestamp":1150416000000},"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                    The Cunningham project seeks to factor numbers of the form\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b Superscript n Baseline plus-or-minus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\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:mo>\n                              \u00b1\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">b^{n}\\pm 1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b equals 2 comma 3 comma ellipsis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>b<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mo>\n                              \u2026\n                              \n                            <\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">b=2,3,\\dots<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    small. One of the most useful techniques is\n                    <italic>Aurifeuillian Factorization<\/italic>\n                    whereby such a number is partially factored by replacing\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"b Superscript n\">\n                        <mml:semantics>\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:annotation encoding=\"application\/x-tex\">b^{n}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    by a polynomial in such a way that polynomial factorization is possible. For example, by substituting\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"y equals 2 Superscript k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\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\">y=2^{k}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    into the polynomial factorization\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis 2 y squared right-parenthesis squared plus 1 equals left-parenthesis 2 y squared minus 2 y plus 1 right-parenthesis left-parenthesis 2 y squared plus 2 y plus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:msup>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:msup>\n                              <mml:mi>y<\/mml:mi>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>2<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msup>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                            <mml:mi>y<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(2y^{2})^{2}+1=(2y^{2}-2y+1)(2y^{2}+2y+1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    we can partially factor\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"2 Superscript 4 k plus 2 Baseline plus 1\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>4<\/mml:mn>\n                                <mml:mi>k<\/mml:mi>\n                                <mml:mo>+<\/mml:mo>\n                                <mml:mn>2<\/mml:mn>\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\">2^{4k+2}+1<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In 1962 Schinzel gave a list of such identities that have proved useful in the Cunningham project; we believe that Schinzel identified\n                    <italic>all<\/italic>\n                    numbers that can be factored by such identities and we prove this if one accepts our definition of what \u201csuch an identity\u201d is. We then develop our theme to similarly factor\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f left-parenthesis b Superscript n Baseline right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>f<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\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:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">f(b^{n})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for any given polynomial\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"f\">\n                        <mml:semantics>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">f<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , using deep results of Faltings from algebraic geometry and Fried from the classification of finite simple groups.\n                  <\/p>","DOI":"10.1090\/s0025-5718-05-01766-7","type":"journal-article","created":{"date-parts":[[2005,11,16]],"date-time":"2005-11-16T10:22:35Z","timestamp":1132136555000},"page":"497-508","source":"Crossref","is-referenced-by-count":6,"title":["Aurifeuillian factorization"],"prefix":"10.1090","volume":"75","author":[{"given":"Andrew","family":"Granville","sequence":"first","affiliation":[]},{"given":"Peter","family":"Pleasants","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2005,6,16]]},"reference":[{"issue":"203","key":"1","doi-asserted-by":"publisher","first-page":"131","DOI":"10.2307\/2152941","article-title":"On computing factors of cyclotomic polynomials","volume":"61","author":"Brent, Richard P.","year":"1993","journal-title":"Math. 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Fried, Applications of the classification of simple groups to monodromy, Part II: Davenport and Hilbert-Siegel problems (to appear)."},{"key":"7","volume-title":"Disquisitiones arithmeticae","author":"Gauss, Carl Friedrich","year":"1966"},{"issue":"3","key":"8","first-page":"501","article-title":"A remark on Aurifeuilian factorizations","volume":"39","author":"Hahn, S.","year":"1994","journal-title":"Math. Japon.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5513","issn-type":"print"},{"key":"9","doi-asserted-by":"publisher","first-page":"416","DOI":"10.2307\/2003131","article-title":"Minimum periods, modulo \ud835\udc5d, of first-order Bell exponential integers","volume":"16","author":"Levine, Jack","year":"1962","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","unstructured":"[Lu] E. Lucas, Th\u00e9or\u00e8mes d\u2019arithm\u00e9tique, Atti. Roy. Acad. Sci. Torino 13 (1878), 271\u2013284."},{"key":"11","doi-asserted-by":"crossref","first-page":"555","DOI":"10.1017\/S0305004100040561","article-title":"On primitive prime factors of \ud835\udc4e\u207f-\ud835\udc4f\u207f","volume":"58","author":"Schinzel, A.","year":"1962","journal-title":"Proc. Cambridge Philos. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0008-1981","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"199","DOI":"10.4064\/aa-31-2-199-204","article-title":"On the equation \ud835\udc66^{\ud835\udc5a}=\ud835\udc43(\ud835\udc65)","volume":"31","author":"Schinzel, A.","year":"1976","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"13","volume-title":"Gesammelte Abhandlungen. 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