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The twenty-second Fermat number is composite

Authors: R. Crandall, J. Doenias, C. Norrie and J. Young
Journal: Math. Comp. 64 (1995), 863-868
MSC: Primary 11Y11; Secondary 11A51
MathSciNet review: 1277765
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Abstract: We have shown by machine proof that $ {F_{22}} = {2^2}^{^{22}} + 1$ is composite. In addition, we reenacted Young and Buell's 1988 resolution of $ {F_{20}}$ as composite, finding agreement with their final Selfridge-Hurwitz residues. We also resolved the character of all extant cofactors of $ {F_n}, n \leq 22$, finding no new primes, and ruling out prime powers.

References [Enhancements On Off] (What's this?)

  • [1] R. Crandall, Projects in scientific computation, Springer-Verlag, Santa Clara, CA, 1994, pp. 145-148. MR 1258083 (95d:65001)
  • [2] R. Crandall and B. Fagin, Discrete weighted transforms and large-integer arithmetic, Math. Comp. 62 (1994), 305-324. MR 1185244 (94c:11123)
  • [3] Wilfrid Keller, Factors of Fermat numbers and large primes of the form $ k{2^n} + 1$, manuscript.
  • [4] A. K. Lenstra, H. W. Lenstra, Jr., M. S. Manasse, and J. M. Pollard, The factorization of the ninth Fermat number, Math. Comp. 61 (1993), 319-349. MR 1182953 (93k:11116)
  • [5] H. Riesel, Prime numbers and computer methods for factorization, Birkhäuser, Boston, 1985, p. 377. MR 897531 (88k:11002)
  • [6] J. L. Selfridge and Alexander Hurwitz, Fermat numbers and Mersenne numbers, Math. Comp. 18 (1964), 146-148. MR 0159775 (28:2991)
  • [7] D. Slowinski, private communication.
  • [8] J. Young and D. Buell, The twentieth Fermat number is composite, Math. Comp. 50 (1988), 261-263. MR 917833 (89b:11012)

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Article copyright: © Copyright 1995 American Mathematical Society

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