This site is supported by donations to The OEIS Foundation.



Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A019434 Fermat primes: primes of form 2^(2^k) + 1, for some k >= 0. 149
3, 5, 17, 257, 65537 (list; graph; refs; listen; history; text; internal format)



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


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


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 BC--2012) 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, avril-juin 2010, pages 30-38. 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


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


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


(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


Cf. A000215, A159611, A220627, A220570.

Sequence in context: A056130 A078726 * A164307 A125045 A093179 A067387

Adjacent sequences:  A019431 A019432 A019433 * A019435 A019436 A019437




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



Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement .

Last modified November 28 21:04 EST 2014. Contains 250406 sequences.