| 释义 |
Fermat NumberA Binomial Number of the form  . The first few for  , 1, 2, ... are 3, 5, 17, 257, 65537,4294967297, ... (Sloane's A000215). The number of Digits for a Fermat number is
Being a Fermat number is the Necessary (but not Sufficient) form a number
 | (2) |
 is to be Prime, then  cannot have any Odd factors  or else  would be a factorable number of the form
 | (3) |
Fermat  conjectured in 1650 that every Fermat number is Prime and Eisenstein (1844) proposed as a problem theproof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88). At present, however, only CompositeFermat numbers  are known for  . An anonymous writer proposed that numbers of the form ,  ,  were Prime. However, this conjecture was refuted when Selfridge (1953) showed that
 | (4) |
 are called generalized Fermat numbers(Ribenboim 1996, pp. 359-360).
Fermat numbers satisfy the Recurrence Relation
 | (5) |
can be shown to be Prime iff it satisfies Pépin's Test
 | (6) |
 | (7) |
In 1770, Euler showed that any Factor of must have the form
 | (8) |
 is a Positive Integer. In 1878, Lucas increased the exponent of 2 by one, showing thatFactors of Fermat numbers must be of the form
 | (9) |
 | (10) |
 (where  is the cofactor to be tested for primality), compute
Then if  , the cofactor is a Probable Prime to the base  ; otherwise is Composite.
In order for a Polygon to be circumscribed about a Circle (i.e., a Constructible Polygon), it must have a number of sides  given by
 | (14) |
 ,  , etc., can be computed in terms of finite numbers of additions, multiplications,and square root extractions iff is of the above form. The only known Fermat Primes are
and it seems unlikely that any more exist.
Factoring Fermat numbers is extremely difficult as a result of their large size. In fact, only  to  have beencomplete factored, as summarized in the following table. Written out explicitly, the complete factorizations are
Here, the final large Prime is not explicitly given since it can be computed by dividing by the other givenfactors.
| Digits | Factors | Digits | Reference | | 5 | 10 | 2 | 3, 7 | Euler 1732 | | 6 | 20 | 2 | 6, 14 | Landry 1880 | | 7 | 39 | 2 | 7, 22 | Morrison and Brillhart 1975 | | 8 | 78 | 2 | 16, 62 | Brent and Pollard 1981 | | 9 | 155 | 3 | 7, 49, 99 | Manasse and Lenstra (In Cipra 1993) | | 10 | 309 | 4 | 8, 10, 40, 252 | Brent 1995 | | 11 | 617 | 5 | 6, 6, 21, 22, 564 | Brent 1988 |
Tables of known factors of Fermat numbers are given by Keller (1983), Brillhart et al. (1988), Young and Buell (1988),Riesel (1994), and Pomerance (1996). Young and Buell (1988) discovered that  is Composite, and Crandall et al. (1995) that  is Composite. A current list of the known factors of Fermat numbers is maintained by Keller, andreproduced in the form of a Mathematica notebook by Weisstein. In these tables, since all factors are ofthe form  , the known factors are expressed in the concise form  . The number of factors for Fermatnumbers for , 1, 2, ... are 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 4, 5, .... See also Cullen Number, Pépin's Test, Pépin's Theorem,Pocklington's Theorem, Polygon, Proth's Theorem, Selfridge-Hurwitz Residue, WoodallNumber
ReferencesBall, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 68-69 and 94-95, 1987.Brent, R. P. ``Factorization of the Eighth Fermat Number.'' Amer. Math. Soc. Abstracts 1, 565, 1980. Brent, R. P. ``Factorisation of F10.'' http://cslab.anu.edu.au/~rpb/F10.html. Brent, R. P ``Factorization of the Tenth Fermat Number.'' Math. Comput. 68, 429-451, 1999. ftp://nimbus.anu.edu.au/pub/Brent/rpb161tr.ps.gz. Brent, R. P. and Pollard, J. M. ``Factorization of the Eighth Fermat Number.'' Math. Comput. 36, 627-630, 1981. Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of  ,  ,  Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc., pp. 1xxxvii and 2-3 of Update 2.2, 1988. Cipra, B. ``Big Number Breakdown.'' Science 248, 1608, 1990. Conway, J. H. and Guy, R. K. ``Fermat's Numbers.'' In The Book of Numbers. New York: Springer-Verlag, pp. 137-141, 1996. Cormack, G. V. and Williams, H. C. ``Some Very Large Primes of the Form  .'' Math. Comput. 35, 1419-1421, 1980. Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 25-26 and 119, 1996. Crandall, R.; Doenias, J.; Norrie, C.; and Young, J. ``The Twenty-Second Fermat Number is Composite.'' Math. Comput. 64, 863-868, 1995. Dickson, L. E. ``Fermat Numbers .'' Ch. 15 in History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, pp. 375-380, 1952. Dixon, R. Mathographics. New York: Dover, p. 53, 1991. Euler, L. ``Observationes de theoremate quodam Fermatiano aliisque ad numeros primos spectantibus.'' Acad. Sci. Petropol. 6, 103-107, ad annos 1732-33 (1738). In Leonhardi Euleri Opera Omnia, Ser. I, Vol. II. Leipzig: Teubner, pp. 1-5, 1915. Gostin, G. B. ``A Factor of  .'' Math. Comput. 35, 975-976, 1980. Gostin, G. B. ``New Factors of Fermat Numbers.'' Math. Comput. 64, 393-395, 1995. Gostin, G. B. and McLaughlin, P. B. Jr. ``Six New Factors of Fermat Numbers.'' Math. Comput. 38, 645-649, 1982. Guy, R. K. ``Mersenne Primes. Repunits. Fermat Numbers. Primes of Shape  .'' §A3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 8-13, 1994. Hallyburton, J. C. Jr. and Brillhart, J. ``Two New Factors of Fermat Numbers.'' Math. Comput. 29, 109-112, 1975. Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 14-15, 1979. Keller, W. ``Factor of Fermat Numbers and Large Primes of the Form  .'' Math. Comput. 41, 661-673, 1983. Keller, W. ``Factors of Fermat Numbers and Large Primes of the Form , II.'' In prep. Keller, W. ``Prime Factors of Fermat Numbers  and Complete Factoring Status.'' http://vamri.xray.ufl.edu/proths/fermat.html. Kraitchik, M. ``Fermat Numbers.'' §3.6 in Mathematical Recreations. New York: W. W. Norton, pp. 73-75, 1942. Landry, F. ``Note sur la décomposition du nombre  (Extrait).'' C. R. Acad. Sci. Paris, 91, 138, 1880. Lenstra, A. K.; Lenstra, H. W. Jr.; Manasse, M. S.; and Pollard, J. M. ``The Factorization of the Ninth Fermat Number.'' Math. Comput. 61, 319-349, 1993. Morrison, M. A. and Brillhart, J. ``A Method of Factoring and the Factorization of  .'' Math. Comput. 29, 183-205, 1975. Pomerance, C. ``A Tale of Two Sieves.'' Not. Amer. Math. Soc. 43, 1473-1485, 1996. Ribenboim, P. ``Fermat Numbers'' and ``Numbers  .'' §2.6 and 5.7 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 83-90 and 355-360, 1996. Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Basel: Birkhäuser, pp. 384-388, 1994. Robinson, R. M. ``A Report on Primes of the Form and on Factors of Fermat Numbers.'' Proc. Amer. Math. Soc. 9, 673-681, 1958. Selfridge, J. L. ``Factors of Fermat Numbers.'' Math. Comput. 7, 274-275, 1953. Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 13 and 78-80, 1993. Sloane, N. J. A. SequenceA000215/M2503in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.  Weisstein, E. W. ``Fermat Numbers.'' Mathematica notebook Fermat.m.
Wrathall, C. P. ``New Factors of Fermat Numbers.'' Math. Comput. 18, 324-325, 1964. Young, J. and Buell, D. A. ``The Twentieth Fermat Number is Composite.'' Math. Comput. 50, 261-263, 1988.
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