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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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112
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OFFSET
| 0,1
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COMMENTS
| 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 (dkamen(AT)rsise.anu.edu.au), Sep 28 2008
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REFERENCES
| 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
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LINKS
| 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
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
| a(n+1) = A180024(A049084(a(n))). [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 08 2010]
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MATHEMATICA
| Select[Table[2^(2^n)+1, {n, 0, 4}], PrimeQ] (from Vladimir Orlovsky (4vladimir(AT) gmail.com), Apr 29 2008)
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PROG
| (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
| Cf. A000215, A159611.
Sequence in context: A056130 A078726 * A164307 A125045 A093179 A067387
Adjacent sequences: A019431 A019432 A019433 * A019435 A019436 A019437
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KEYWORD
| nonn,hard,nice,more
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), David W. Wilson (davidwwilson(AT)comcast.net)
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