{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:18:50Z","timestamp":1776838730576,"version":"3.51.2"},"reference-count":13,"publisher":"American Mathematical Society (AMS)","issue":"213","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We show that the minimum period modulo\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                    of the Bell exponential integers is\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis p Superscript p Baseline minus 1 right-parenthesis slash left-parenthesis p minus 1 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>p<\/mml:mi>\n                              <mml:mi>p<\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mo>\/<\/mml:mo>\n                            <\/mml:mrow>\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:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">(p^p-1)\/(p-1)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for all primes\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p greater-than 102\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>102<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p&gt;102<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and several larger\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                    . Our proof of this result requires the prime factorization of these periods. For some primes\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                    the factoring is aided by an algebraic formula called an Aurifeuillian factorization. We explain how the coefficients of the factors in these formulas may be computed.\n                  <\/p>","DOI":"10.1090\/s0025-5718-96-00683-7","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"383-391","source":"Crossref","is-referenced-by-count":9,"title":["Aurifeuillian factorizations and the period of the Bell numbers modulo a prime"],"prefix":"10.1090","volume":"65","author":[{"suffix":"Jr.","given":"Samuel","family":"Wagstaff","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1996]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"H. W. Becker and D. H. Browne,  Problem E461 and solution, Amer. Math. 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(2) 45 (1915), 49\u201375.","DOI":"10.1017\/S0021859600002422"},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"649","DOI":"10.2307\/1971363","article-title":"Factoring integers with elliptic curves","volume":"126","author":"Lenstra, H. W., Jr.","year":"1987","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"},{"key":"6","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":"7","unstructured":"E. Lucas, Th\u00e9or\u00e8ms d\u2019arithm\u00e9tique, Atti R. Accad. Sci. Torino 13 (1877\u201378), 271\u2013284."},{"key":"8","unstructured":"E. Lucas, Sur la s\u00e9rie r\u00e9currente de Fermat, Bull. Bibl. Storia Sc. Mat. e Fis. 11 (1878), 783\u2013789."},{"key":"9","series-title":"Lecture Notes in Computer Science","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/3-540-39757-4","volume-title":"Advances in cryptology","volume":"209","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/3540160760"},{"key":"10","series-title":"Progress in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1089-2","volume-title":"Prime numbers and computer methods for factorization","volume":"57","author":"Riesel, Hans","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0817632913"},{"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"},{"key":"12","unstructured":"J. Touchard, Propri\u00e9t\u00e9s arithm\u00e9tiques de certains nombres recurrents, Ann. Soc. Sci. Bruxelles 53A (1933), 21\u201331."},{"key":"13","doi-asserted-by":"publisher","first-page":"537","DOI":"10.2307\/1968938","article-title":"On polynomials with only real roots","volume":"40","author":"Erd\u00f6s, P.","year":"1939","journal-title":"Ann. of Math. 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