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A019434 Fermat primes: primes of the form 2^(2^k) + 1, for some k >= 0. 236
3, 5, 17, 257, 65537 (list; graph; refs; listen; history; text; internal format)
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

Numbers n > 1 such that n * 2^(n-2) divides (n-1)! + 2^(n-1). - Thomas Ordowski, Jan 15 2015

From Jaroslav Krizek, Mar 17 2016: (Start)

Primes p such that phi(p) = 2*phi(p-1); primes from A171271.

Primes p such that sigma(p-1) = 2p - 3.

Primes p such that sigma(p-1) = 2*sigma(p) - 5.

For n > 1, a(n) = primes p such that p = 4 * phi((p-1) / 2) + 1.

Subsequence of A256444 and A256439.

Conjectures:

1) primes p such that phi(p) = 2*phi(p-2).

2) primes p such that phi(p) = 2*phi(p-1) = 2*phi(p-2).

3) primes p such that p = sigma(phi(p-2)) + 2.

4) primes p such that phi(p-1) + 1 divides p + 1.

5) numbers n such that sigma(n-1) = 2*sigma(n) - 5. (End)

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 a heuristic argument that this sequence is finite. - T. D. Noe, Jun 14 2010

LINKS

Table of n, a(n) for n=1..5.

Cyril Banderier, Pepin's Criterion For Fermat Numbers (in French)

Kent D. Boklan, John H. Conway, Expect at most one billionth of a new Fermat Prime!, arXiv:1605.01371 [math.NT], 2016.

P. Bruillard, S.-H. Ng, E. Rowell, Z. Wang, On modular categories, arXiv preprint arXiv:1310.7050 [math.QA], 2013-2015.

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 [math.HO], 2012. - 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

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: A078726 * A164307 A262534 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

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Last modified December 4 15:05 EST 2016. Contains 278750 sequences.