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A019434
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Fermat primes: primes of form 2^(2^k) + 1, for some k >= 0.
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122
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OFFSET
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1,1
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COMMENTS
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It is conjectured that there are only 5 terms. Currently it has been shown that 2^(2^k) + 1 is composite for 5<=k<=32 (see Eric Weisstein's Fermat Primes link). - Dmitry Kamenetsky, Sep 28 2008
No Fermat prime is a Brazilian number. Hence, Fermat primes belong to A220627. For a proof, see Proposition 3 page 36 on "Les nombres brésiliens" in Links. - Bernard Schott, Dec 29 2012
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REFERENCES
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G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see Table 1, p. 458.
R. K. Guy, Unsolved Problems in Number Theory, A3.
Hardy and Wright, An Introduction to the Theory of Numbers, bottom of page 18 in the sixth edition, gives heuristic argument that sequence is finite. - T. D. Noe, Jun 14 2010
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, Arxiv preprint arXiv:1202.3670, 2012 - From N. J. A. Sloane, Jun 13 2012
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LINKS
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Table of n, a(n) for n=1..5.
C. Banderier, Pepin's Criterion For Fermat Numbers
C. K. Caldwell, The Prime Glossary, Fermat number
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38. Local copy, included here with permission from the editors of Quadrature.
Eric Weisstein's World of Mathematics, Pepin's Test
Eric Weisstein's World of Mathematics, Fermat Number
Eric Weisstein's World of Mathematics, Fermat Prime
Eric Weisstein's World of Mathematics, Pepins Test
Wikipedia, Fermat prime
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FORMULA
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a(n+1) = A180024(A049084(a(n))). - Reinhard Zumkeller, Aug 08 2010
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MATHEMATICA
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Select[Table[2^(2^n)+1, {n, 0, 4}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
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PROG
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(MAGMA) [2^(2^n)+1: n in [0..4]|IsPrime(2^(2^n)+1)] [Arkadiusz Wesolowski, Jun 09 2011]
(PARI) for(i=0, 10, isprime(2^2^i+1) & print1(2^2^i+1, ", ")) \\ - M. F. Hasler, Nov 21 2009
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CROSSREFS
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Cf. A000215, A159611, A220627, A220570.
Sequence in context: A056130 A078726 * A164307 A125045 A093179 A067387
Adjacent sequences: A019431 A019432 A019433 * A019435 A019436 A019437
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KEYWORD
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nonn,hard,nice,more
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AUTHOR
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N. J. A. Sloane, David W. Wilson
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STATUS
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approved
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