The twenty-fourth Fermat number is composite

Abstract

We have shown by machine proof that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 24 equals 2 Superscript 2 Super Superscript 24 Baseline plus 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>24</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">F_{24} = 2^{2^{24}} + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 24"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">F_{24}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 24"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>24</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">F_{24}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 23"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>23</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">F_{23}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and via the Suyama test determined that the known cofactor of this number is composite.