@article{CMP2003,
author={R. E. Crandall and E. W. Mayer and J. S. Papadopoulos},
title={The twenty-fourth {Fermat} number is composite},
abstract={We have shown by machine proof that $F_{24} = 2^{2^{24}} + 1$ is composite.
The rigorous P{\'e}pin 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 P{\'e}pin 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 P{\'e}pin chain. We deposited
a final P{\'e}pin 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 P{\'e}pin residue for $F_{23}$, and via
the Suyama test determined that the known cofactor of this number is composite.},
journal= MC,
volume={72},
year={2003},
pages={1555--1572}
}
