A092506 Prime numbers of the form 2^n + 1.

2, 3, 5, 17, 257, 65537

Offset: 1

Author

Jorge Coveiro, Apr 05 2004

Keywords

• nonn
• hard

Comments

2 together with the Fermat primes A019434.

Obviously if 2^n + 1 is a prime then either n = 0 or n is a power of 2. - N. J. A. Sloane, Apr 07 2004

Numbers m > 1 such that 2^(m-2) divides (m-1)! and m divides (m-1)! + 1. - Thomas Ordowski, Nov 25 2014

From Jaroslav Krizek, Mar 06 2016: (Start)

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

Also primes of the form 2^n + 3*(-1)^n - 2 for n >= 0 because for odd n, 2^n - 5 is divisible by 3.

Also primes of the form 2^n + 6*(-1)^n - 5 for n >= 0 because for odd n, 2^n - 11 is divisible by 3.

Also primes of the form 2^n + 15*(-1)^n - 14 for n >= 0 because for odd n, 2^n - 29 is divisible by 3. (End)

Exactly the set of primes p such that any number congruent to a primitive root (mod p) must have at least one prime divisor that is also congruent to a primitive root (mod p). See the links for a proof. - Rafay A. Ashary, Oct 13 2016

Conjecture: these are the only solutions to the equation A000010(x)+A000010(x-1)=floor((3x-2)/2). - Benoit Cloitre, Mar 02 2018

For n > 1, if 2^n + 1 divides 3^(2^(n-1)) + 1, then 2^n + 1 is a prime. - Jinyuan Wang, Oct 13 2018

The prime numbers occurring in A003401. Also, the prime numbers dividing at least one term of A003401. - Jeppe Stig Nielsen, Jul 24 2019

Links

• Rafay A. Ashary, A Property of A092506
• Barry Brent, On the constant terms of certain meromorphic modular forms for Hecke groups, arXiv:2212.12515 [math.NT], 2022.
• Eric Weisstein's World of Mathematics, Fermat Prime
• Eric Weisstein's World of Mathematics, Fermat Number

Crossrefs

A019434 is the main entry for these numbers.

Cf. A000010, A000040, A000051, A000215, A003401.

Programs

GAP

Filtered(List([1..20],n->2^n+1),IsPrime); # Muniru A Asiru, Oct 25 2018

Magma

[2^n + 1 : n in [0..25] | IsPrime(2^n+1)]; // Vincenzo Librandi, Oct 14 2018

Mathematica

Select[2^Range[0,100]+1,PrimeQ] (* Harvey P. Dale, Aug 02 2015 *)

PARI

print1(2); for(n=0,9, if(ispseudoprime(t=2^2^n+1), print1(", "t))) \\ Charles R Greathouse IV, Aug 29 2016