A277967 Number of even numbers b with 0 < b < 2^n such that b^(2^n) + 1 is prime.

0, 1, 2, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 2, 3, 4, 1

Offset

1,3

Comments

The choice whether to take b < 2^n or b <= 2^n matters only for n=1 and n=2 unless there are more primes like 2^2+1 and 4^4+1 (see A121270).

Perfect squares b are allowed.

a(20) was determined after a lengthy computation by distributed project PrimeGrid, cf. A321323. - Jeppe Stig Nielsen, Jan 02 2019

Links

Table of n, a(n) for n=1..20.

Y. Gallot, Status of the smallest base values yielding Generalized Fermat primes.

PrimeGrid, GFN Prime Search Status and History.

Example

For n=18, we get b^262144 + 1 is prime for b=24518, 40734, 145310, 361658, 525094, ...; the first 3 of these b values are strictly below 262144, hence a(18)=3.
The corresponding primes are 2^4+1; 2^8+1, 4^8+1; 2^16+1; 30^32+1; 120^128+1; 46^512+1; etc.

Mathematica

Table[Count[Range[2, 2^n - 1, 2], b_ /; PrimeQ[b^(2^n) + 1]], {n, 9}] (* Michael De Vlieger, Nov 10 2016 *)

Prog

(PARI) a(n)=sum(k=1, 2^(n-1)-1, ispseudoprime((2*k)^2^n+1)) \\ slow, only probabilistic primality test

Crossrefs

Cf. A056993, A121270, A259835, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A226528, A226529, A226530, A251597, A253854, A244150, A243959, A321323, etc.

Sequence in context: A259287 A342592 A090464 * A196049 A194821 A044934

Adjacent sequences: A277964 A277965 A277966 * A277968 A277969 A277970

Keyword

nonn,hard,more

Author

Jeppe Stig Nielsen, Nov 06 2016

Extensions

a(20) from Jeppe Stig Nielsen, Jan 02 2019

Status

approved