A019434 - OEIS (original) (raw)

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^(n-2) divides (n-1)! + 2^(n-1). - Thomas Ordowski, Jan 15 2015

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.

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)

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. - Juri-Stepan 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. - Juri-Stepan 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

If there are only 5 Fermat primes, then there are only 31 odd order groups which have a 2-group automorphism group. See the Miles Englezou link for a proof. - Miles Englezou, Mar 10 2025

REFERENCES

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 137-141, 197.

John Derbyshire, Unknown Quantity: A Real and Imaginary History of Algebra. Washington, DC: Joseph Henry Press (2006): 112-113.

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 268.

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.

Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.2 Prime Numbers, pp. 78-79.

Richard 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.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 7, 70.

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 136-137.

LINKS

Chris K. Caldwell, The Prime Glossary, Fermat number.

Salah Eddine Rihane, Chèfiath Awero Adegbindin, and Alain Togbé, Fermat Padovan And Perrin Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.6.2.

Bernard Schott, Les nombres brésiliens, Quadrature (avril-juin 2010) No. 76, 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, Pépin's Test.

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

(SageMath) [2^(2^n)+1 for n in (0..4) if is_prime(2^(2^n)+1)] # G. C. Greubel, Mar 07 2019