

A019434


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


351




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. So Fermat primes belong to A220627. For proof see Proposition 3 page 36 in "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^(n2) divides (n1)! + 2^(n1).  Thomas Ordowski, Jan 15 2015
Primes p such that phi(p) = 2*phi(p1); primes from A171271.
Primes p such that sigma(p1) = 2p  3.
Primes p such that sigma(p1) = 2*sigma(p)  5.
For n > 1, a(n) = primes p such that p = 4 * phi((p1) / 2) + 1.
Conjectures:
1) primes p such that phi(p) = 2*phi(p2).
2) primes p such that phi(p) = 2*phi(p1) = 2*phi(p2).
3) primes p such that p = sigma(phi(p2)) + 2.
4) primes p such that phi(p1) + 1 divides p + 1.
5) numbers n such that sigma(n1) = 2*sigma(n)  5. (End)
Odd primes p such that ratio of the form (the number of nonnegative m < p such that m^q == m (mod p))/(the number of nonnegative m < p such that m^q == m (mod p)) is a divisor of p for all nonnegative q.  JuriStepan Gerasimov, Oct 13 2020
Numbers n such that tau(n)*(number of distinct ratio (the number of nonnegative m < n such that m^q == m (mod n))/(the number of nonnegative m < n such that m^q == m (mod n))) for nonnegative q is equal to 4.  JuriStepan Gerasimov, Oct 22 2020
The numbers of primitive roots for the five known terms are 1, 2, 8, 128, 32768.  Gary W. Adamson, Jan 13 2022
Prime numbers such that every residue is either a primitive root or a quadratic residue.  Keith Backman, Jul 11 2022


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 an heuristic argument that this sequence is finite.


LINKS

P. Bruillard, S.H. Ng, E. Rowell, and Z. Wang, On modular categories, arXiv:1310.7050 [math.QA], 20132015.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avriljuin 2010, pages 3038. Local copy, included here with permission from the editors of Quadrature.


FORMULA



MATHEMATICA



PROG

(PARI) for(i=0, 10, isprime(2^2^i+1) && print1(2^2^i+1, ", ")) \\ M. F. Hasler, Nov 21 2009
(Sage) [2^(2^n)+1 for n in (0..4) if is_prime(2^(2^n)+1)] # G. C. Greubel, Mar 07 2019


CROSSREFS



KEYWORD

nonn,hard,nice


AUTHOR



STATUS

approved



