

A019434


Fermat primes: primes of form 2^(2^k) + 1, for some k >= 0.


149




OFFSET

1,1


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, 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
This sequence and A001220 are disjoint (see "Other theorems about Fermat numbers" in Wikipedia link).  Felix Fröhlich, Sep 07 2014


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


LINKS

Table of n, a(n) for n=1..5.
C. Banderier, Pepin's Criterion For Fermat Numbers
P. Bruillard, S.H. Ng, E. Rowell, Z. Wang, On modular categories, arXiv preprint arXiv:1310.7050, 2013
C. K. Caldwell, The Prime Glossary, Fermat number
Wilfrid Keller, Prime factors k.2^n + 1 of Fermat numbers F_m
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC2012) and another new proof, arXiv preprint arXiv:1202.3670, 2012  From N. J. A. Sloane, Jun 13 2012
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avriljuin 2010, pages 3038. Local copy, included here with permission from the editors of Quadrature.
Eric Weisstein's World of Mathematics, Fermat Number
Eric Weisstein's World of Mathematics, Fermat Prime
Eric Weisstein's World of Mathematics, Pepin's Test
Wikipedia, Fermat number


FORMULA

a(n+1) = A180024(A049084(a(n))).  Reinhard Zumkeller, Aug 08 2010


MATHEMATICA

Select[Table[2^(2^n)+1, {n, 0, 4}], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)


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


CROSSREFS

Cf. A000215, A159611, A220627, A220570.
Sequence in context: A056130 A078726 * A164307 A125045 A093179 A067387
Adjacent sequences: A019431 A019432 A019433 * A019435 A019436 A019437


KEYWORD

nonn,hard,nice,more


AUTHOR

N. J. A. Sloane, David W. Wilson


STATUS

approved



