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Mathematics of Computation

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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
Published electronically: December 6, 2002
MathSciNet review: 1972753
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Abstract: We have shown by machine proof that $F_{24} = 2^{2^{24}} + 1$ 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 $F_{24}$. 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 $F_{24}$ 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 $F_{23}$, and via the Suyama test determined that the known cofactor of this number is composite.

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Additional Information

Richard E. Crandall
Affiliation: Center for Advanced Computation, Reed College, Portland, Oregon 97202

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

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

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