{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:47:04Z","timestamp":1776728824931,"version":"3.51.2"},"reference-count":32,"publisher":"American Mathematical Society (AMS)","issue":"243","license":[{"start":{"date-parts":[[2003,12,6]],"date-time":"2003-12-06T00:00:00Z","timestamp":1070668800000},"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":"

\n We have shown by machine proof that\n \n \n \n \n \n F<\/mml:mi>\n \n 24<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n =<\/mml:mo>\n \n 2<\/mml:mn>\n \n \n 2<\/mml:mn>\n \n 24<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:msup>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n F_{24} = 2^{2^{24}} + 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is composite. The rigorous P\u00e9pin primality test was performed using independently developed programs running simultaneously on two different, physically separated processors. Each program employed a floating-point, FFT-based discrete weighted transform (DWT) to effect multiplication modulo\n \n \n \n \n F<\/mml:mi>\n \n 24<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n F_{24}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The final, respective P\u00e9pin residues obtained by these two machines were in complete agreement. Using intermediate residues stored periodically during one of the floating-point runs, a separate algorithm for pure-integer negacyclic convolution verified the result in a \u201cwavefront\u201d paradigm, by running simultaneously on numerous additional machines, to effect piecewise verification of a saturating set of deterministic links for the P\u00e9pin chain. We deposited a final P\u00e9pin residue for possible use by future investigators in the event that a proper factor of\n \n \n \n \n F<\/mml:mi>\n \n 24<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n F_{24}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n should be discovered; herein we report the more compact, traditional Selfridge-Hurwitz residues. For the sake of completeness, we also generated a P\u00e9pin residue for\n \n \n \n \n F<\/mml:mi>\n \n 23<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msub>\n F_{23}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and via the Suyama test determined that the known cofactor of this number is composite.\n <\/p>","DOI":"10.1090\/s0025-5718-02-01479-5","type":"journal-article","created":{"date-parts":[[2003,4,18]],"date-time":"2003-04-18T13:09:02Z","timestamp":1050671342000},"page":"1555-1572","source":"Crossref","is-referenced-by-count":11,"title":["The twenty-fourth Fermat number is composite"],"prefix":"10.1090","volume":"72","author":[{"given":"Richard","family":"Crandall","sequence":"first","affiliation":[]},{"given":"Ernst","family":"Mayer","sequence":"additional","affiliation":[]},{"given":"Jason","family":"Papadopoulos","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,12,6]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"R. C. Agarwal and J. W. Cooley,\u201cFourier Transform and Convolution Subroutines for the IBM 3090 Vector Facility,\u201d IBM Journal of Research and Development 30 (1986), 145 - 162.","DOI":"10.1147\/rd.302.0145"},{"key":"2","doi-asserted-by":"crossref","unstructured":"M. Ashworth and A. G. Lyne, \u201cA Segmented FFT Algorithm for Vector Computers,\u201d Parallel Computing 6 (1988), 217-224.","DOI":"10.1016\/0167-8191(88)90086-5"},{"key":"3","doi-asserted-by":"crossref","unstructured":"D. Bailey, \u201cFFTs in External or Hierarchical Memory,\" (1989) manuscript.","DOI":"10.1145\/76263.76288"},{"issue":"203","key":"4","doi-asserted-by":"publisher","first-page":"151","DOI":"10.2307\/2152942","article-title":"Irregular primes and cyclotomic invariants to four million","volume":"61","author":"Buhler, J.","year":"1993","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"200","key":"5","doi-asserted-by":"publisher","first-page":"717","DOI":"10.2307\/2153086","article-title":"Irregular primes to one million","volume":"59","author":"Buhler, J. P.","year":"1992","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"6","first-page":"318","article-title":"Sur le proc\u00e9d\u00e9 d\u2019interpolation de Fej\u00e9r","volume":"24","author":"Losinsky, S.","year":"1939","journal-title":"C. R. (Doklady) Acad. Sci. URSS (N.S.)"},{"key":"7","unstructured":"C. Burrus, DFT\/FFT and Convolution Algorithms: Theory and Implementation, Wiley, New York, 1985."},{"key":"8","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":"210","key":"9","doi-asserted-by":"publisher","first-page":"863","DOI":"10.2307\/2153456","article-title":"The twenty-second Fermat number is composite","volume":"64","author":"Crandall, R.","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"205","key":"10","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":"11","unstructured":"R. Crandall, \u201cParallelization of Pollard-rho factorization,\" manuscript, http:\/\/www.perfsci.com, (1999)."},{"key":"12","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-9316-0","volume-title":"Prime numbers","author":"Crandall, Richard","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0387947779"},{"key":"13","volume-title":"An introduction to probability theory and its applications. Vol. I","author":"Feller, William","year":"1968","edition":"3"},{"key":"14","unstructured":"M. Frigo and S. Johnson, \u201cThe fastest Fourier transform in the west,\" http:\/\/theory.lcs.mit.edu\/fftw."},{"key":"15","unstructured":"E. W. Mayer, GIMPS Source Code Timings Page, http:\/\/hogranch.com\/mayer\/gimps_timings.html#accuracy."},{"key":"16","unstructured":"G. B. Valor, private communication (2001)."},{"key":"17","unstructured":"GIMPS homepage, http:\/\/www.mersenne.org."},{"key":"18","unstructured":"IEEE Standard for Binary Floating-Point Arithmetic, ANSI\/IEEE Standard 754-1985, IEEE (1985)."},{"key":"19","unstructured":"W. Keller, Fermat-number website data: http:\/\/www.prothsearch.net\/fermat.html."},{"key":"20","unstructured":"W. Keller, private communication (1999)."},{"key":"21","unstructured":"H. Lenstra, private communication (1999)."},{"key":"22","unstructured":"E. Mayer, Mlucas: an open-source program for testing the character of Mersenne numbers. http:\/\/hogranch.com\/mayer\/README.html."},{"key":"23","doi-asserted-by":"crossref","unstructured":"R. R. Schaller, Moore\u2019s law: past, present and future, IEEE Spectrum 34 (1997), 52-59. (Also cf. http:\/\/www.intel.com\/research\/silicon\/mooreslaw.htm.)","DOI":"10.1109\/6.591665"},{"key":"24","series-title":"Springer Series in Information Sciences","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-00551-4","volume-title":"Fast Fourier transform and convolution algorithms","volume":"2","author":"Nussbaumer, Henri J.","year":"1981","ISBN":"https:\/\/id.crossref.org\/isbn\/3540101594"},{"key":"25","unstructured":"C. Percival, PiHex: A distributed effort to calculate Pi. http:\/\/www.cecm.sfu.ca\/projects\/pihex\/index.html."},{"key":"26","series-title":"Progress in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-0251-6","volume-title":"Prime numbers and computer methods for factorization","volume":"126","author":"Riesel, Hans","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0817637435","edition":"2"},{"key":"27","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1007\/bf02242355","article-title":"Schnelle Multiplikation grosser Zahlen","volume":"7","author":"Sch\u00f6nhage, A.","year":"1971","journal-title":"Computing (Arch. Elektron. Rechnen)","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"key":"28","doi-asserted-by":"publisher","first-page":"146","DOI":"10.2307\/2003419","article-title":"Fermat numbers and Mersenne numbers","volume":"18","author":"Selfridge, J. L.","year":"1964","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"29","doi-asserted-by":"crossref","unstructured":"V. Trevisan and J. B. Carvalho, \"The composite character of the twenty-second Fermat number,\" J. Supercomputing 9 (1995), 179-182.","DOI":"10.1007\/BF01245403"},{"key":"30","unstructured":"G. Woltman, private communication (1999)."},{"issue":"181","key":"31","doi-asserted-by":"publisher","first-page":"261","DOI":"10.2307\/2007929","article-title":"The twentieth Fermat number is composite","volume":"50","author":"Young, Jeff","year":"1988","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"32","unstructured":"P. Zimmerman, private communication (2001)."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-243\/S0025-5718-02-01479-5\/S0025-5718-02-01479-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-243\/S0025-5718-02-01479-5\/S0025-5718-02-01479-5.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:21:08Z","timestamp":1776727268000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-243\/S0025-5718-02-01479-5\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,12,6]]},"references-count":32,"journal-issue":{"issue":"243","published-print":{"date-parts":[[2003,7]]}},"alternative-id":["S0025-5718-02-01479-5"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-02-01479-5","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":[[2002,12,6]]}}}