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CMP2003

R. E. Crandall, E. W. Mayer and J. S. Papadopoulos, "The twenty-fourth Fermat number is composite," Math. Comp., 72 (2003) 1555--1572.

Abstract: We have shown by machine proof that F₂₄ = 2^{2²⁴} + 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.