The twenty-fourth fermat number is composite

Authors:
Richard E. Crandall;Ernst W. Mayer;Jason S. Papadopoulos
Affiliations:
Center for Advanced Computation, Reed College, Portland, Oregon;Department of Mech. & Aerospace Engineering, Case Western Reserve University, Cleveland, Ohio and 10190 Parkwood Dr. Apt. 1, Cupertino, CA;Department of Elec. & Comp. Engineering, University of Maryland, College Park, Maryland
Venue:
Mathematics of Computation
Year:
2003

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Abstract

We have shown by machine proof that F24 = 2224 + 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 F24. 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 F24 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 F23, and via the Suyama test determined that the known cofactor of this number is composite.