The twenty-second Fermat number is composite
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- by R. Crandall, J. Doenias, C. Norrie and J. Young PDF
- Math. Comp. 64 (1995), 863-868 Request permission
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
- Richard E. Crandall, Projects in scientific computation, TELOS. The Electronic Library of Science, Santa Clara, CA; Springer-Verlag, New York, 1994. With one Macintosh/IBM-PC floppy disk (3.5 inch). MR 1258083, DOI 10.1007/978-1-4612-4324-3
- Richard Crandall and Barry Fagin, Discrete weighted transforms and large-integer arithmetic, Math. Comp. 62 (1994), no. 205, 305–324. MR 1185244, DOI 10.1090/S0025-5718-1994-1185244-1 Wilfrid Keller, Factors of Fermat numbers and large primes of the form $k{2^n} + 1$, manuscript.
- 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), no. 203, 319–349. MR 1182953, DOI 10.1090/S0025-5718-1993-1182953-4
- Hans Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, vol. 57, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 897531, DOI 10.1007/978-1-4757-1089-2
- J. L. Selfridge and Alexander Hurwitz, Fermat numbers and Mersenne numbers, Math. Comp. 18 (1964), 146–148. MR 159775, DOI 10.1090/S0025-5718-1964-0159775-8 D. Slowinski, private communication.
- Jeff Young and Duncan A. Buell, The twentieth Fermat number is composite, Math. Comp. 50 (1988), no. 181, 261–263. MR 917833, DOI 10.1090/S0025-5718-1988-0917833-8
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Math. Comp. 64 (1995), 863-868
- MSC: Primary 11Y11; Secondary 11A51
- DOI: https://doi.org/10.1090/S0025-5718-1995-1277765-9
- MathSciNet review: 1277765