The twenty-fourth Fermat number is composite

Authors:
Richard E. Crandall, Ernst W. Mayer and Jason S. Papadopoulos

Journal:
Math. Comp. **72** (2003), 1555-1572

MSC (2000):
Primary 11Y11, 11Y16, 68Q25, 11A51

DOI:
https://doi.org/10.1090/S0025-5718-02-01479-5

Published electronically:
December 6, 2002

MathSciNet review:
1972753

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We have shown by machine proof that is composite. The rigorous Pépin 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 . The final, respective Pépin 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 ``wavefront'' paradigm, by running simultaneously on numerous additional machines, to effect piecewise verification of a saturating set of deterministic links for the Pépin chain. We deposited a final Pépin residue for possible use by future investigators in the event that a proper factor of should be discovered; herein we report the more compact, traditional Selfridge-Hurwitz residues. For the sake of completeness, we also generated a Pépin residue for , and via the Suyama test determined that the known cofactor of this number is composite.

**1.**R. C. Agarwal and J. W. Cooley,``Fourier Transform and Convolution Subroutines for the IBM 3090 Vector Facility,''*IBM Journal of Research and Development***30**(1986), 145 - 162. CMP**18:12****2.**M. Ashworth and A. G. Lyne, ``A Segmented FFT Algorithm for Vector Computers,''*Parallel Computing***6**(1988), 217-224. CMP**20:07****3.**D. Bailey, ``FFTs in External or Hierarchical Memory," (1989) manuscript.**4.**J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla, ``Irregular Primes to Four Million,''*Math. Comp.***61**, 151-153, (1993). MR**93k:11014****5.**J. Buhler, R. Crandall, R. W. Sompolski, ``Irregular Primes to One Million,''*Math. Comp.***59**, 717-722 (1992). MR**93a:11106****6.**J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla, A. Shokrollahi, ``Irregular Primes and Cyclotomic Invariants to Eight Million,'' manuscript (1996).^{}**7.**C. Burrus,*DFT/FFT and Convolution Algorithms: Theory and Implementation*, Wiley, New York, 1985.**8.**R. E. Crandall,*Topics in Advanced Scientific Computation*, Springer, New York, 1996. MR**97g:65005****9.**R. Crandall, J. Doenias, C. Norrie, and J. Young, ``The Twenty-Second Fermat Number is Composite,"*Math. Comp.***64**(1995), 863-868. MR**95f:11104****10.**R. Crandall and B. Fagin, ``Discrete Weighted Transforms and Large-Integer Arithmetic,"*Math. Comp.***62**(1994), 305-324. MR**94c:11123****11.**R. Crandall, ``Parallelization of Pollard-rho factorization," manuscript,`http://www.perfsci.com`, (1999).**12.**R. Crandall and C. Pomerance,*Prime numbers: a computational perspective*, Springer, New York, (2001) MR**2002a:11007****13.**W. Feller, ``An Introduction to Probability Theory and Its Applications," Vol. I, 3rd ed., Wiley, New York, 1968. MR**37:3604****14.**M. Frigo and S. Johnson, ``The fastest Fourier transform in the west,"`http://theory.lcs.mit.edu/fftw`.**15.**E. W. Mayer, GIMPS Source Code Timings Page,`http://hogranch.com/mayer/gimps_timings.html#accuracy`.**16.**G. B. Valor, private communication (2001).**17.**GIMPS homepage,`http://www.mersenne.org`.**18.***IEEE Standard for Binary Floating-Point Arithmetic*, ANSI/IEEE Standard 754-1985, IEEE (1985).**19.**W. Keller, Fermat-number website data:`http://www.prothsearch.net/fermat.html`.**20.**W. Keller, private communication (1999).**21.**H. Lenstra, private communication (1999).**22.**E. Mayer, Mlucas: an open-source program for testing the character of Mersenne numbers.`http://hogranch.com/mayer/README.html`.**23.**R. R. Schaller,*Moore's law: past, present and future*,*IEEE Spectrum***34**(1997), 52-59. (Also cf.`http://www.intel.com/research/silicon/mooreslaw.htm`.)**24.**H. J. Nussbaumer,*Fast Fourier Transform and Convolution Algorithms*, 2nd ed., Volume 2 of Springer Series in Information Sciences, Springer, New York, 1982. MR**83e:65219****25.**C. Percival, PiHex: A distributed effort to calculate Pi.`http://www.cecm.sfu.ca/projects/pihex/index.html`.**26.**H. Riesel,*Prime Numbers and Computer Methods for Factorization*, 2nd ed., Birkhäuser, Boston, 1994. MR**95h:11142****27.**A. Schönhage 1971, Schnelle Multiplikation grosser Zahlen,*Computing***7**(1971) 282-292. MR**45:1431****28.**J. L. Selfridge and A. Hurwitz, ``Fermat numbers and Mersenne numbers,"*Math. Comp.***18**(1964), 146-148. MR**28:2991****29.**V. Trevisan and J. B. Carvalho, "The composite character of the twenty-second Fermat number,"*J. Supercomputing***9**(1995), 179-182.**30.**G. Woltman, private communication (1999).**31.**J. Young amd D. Buell, ``The Twentieth Fermat Number is Composite,"*Math. Comp.***50**(1988), 261-263. MR**89b:11012****32.**P. Zimmerman, private communication (2001).

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
11Y11,
11Y16,
68Q25,
11A51

Retrieve articles in all journals with MSC (2000): 11Y11, 11Y16, 68Q25, 11A51

Additional Information

**Richard E. Crandall**

Affiliation:
Center for Advanced Computation, Reed College, Portland, Oregon 97202

Email:
crandall@reed.edu

**Ernst W. Mayer**

Affiliation:
Department of Mech. & Aerospace Engineering, Case Western Reserve University, Cleveland, Ohio 44106

Address at time of publication:
10190 Parkwood Dr. Apt. 1, Cupertino, CA 95014

Email:
ewmayer@aol.com

**Jason S. Papadopoulos**

Affiliation:
Department of Elec. & Comp. Engineering, University of Maryland, College Park, Maryland 20742

Email:
jasonp@boo.net

DOI:
https://doi.org/10.1090/S0025-5718-02-01479-5

Received by editor(s):
October 14, 1999

Received by editor(s) in revised form:
September 5, 2001

Published electronically:
December 6, 2002

Article copyright:
© Copyright 2002
American Mathematical Society