A340480 Primes p such that (p^8 + 1)/2 is prime.

13, 43, 47, 53, 239, 373, 409, 433, 491, 557, 577, 859, 1021, 1103, 1307, 1531, 1699, 1753, 1777, 1871, 2053, 2083, 2297, 2467, 2503, 2593, 2797, 2957, 3251, 3307, 3323, 3511, 3613, 4099, 4523, 4637, 4951, 4999, 5591, 5657, 5693, 5801, 5827, 5849, 6043, 6163

Offset: 1

Author

Jon E. Schoenfield, Jan 31 2021

Keywords

• nonn

Comments

Expressions of the form m^j + 1 can be factored (e.g., m^3 + 1 = (m + 1)*(m^2 - m + 1)) for any positive integer j except when j is a power of 2, so (p^j + 1)/2 for prime p cannot be prime unless j is a power of 2. A005383, A048161, A176116, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, j=2^2=4, and j=2^3=8, respectively.

(p^8 + 1)/2 is divisible by 17 when m mod 34 is 3, 5, 7, 11, 23, 27, 29, or 31.

Examples

			(3^8 + 1)/2 = 3281 = 17*193, so 3 is not a term.
(13^8 + 1)/2 = 407865361 is prime, so 13 is a term.
(17^8 + 1)/2 = 3487878721 = 18913 * 184417, so 17 is not a term.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Crossrefs

Primes p such that (p^(2^k) + 1)/2 is prime: A005383 (k=0), A048161 (k=1), A176116 (k=2), (this sequence) (k=3).

Programs

Mathematica

Prime[Position[Table[(Prime[p]^8 + 1)/2, {p, 1, 803}], Integer?PrimeQ]] // Flatten (* _Robert P. P. McKone, Jan 31 2021 *)

PARI

isok(p) = isprime(p) && (p>2) && isprime((p^8 + 1)/2); \\ Michel Marcus, Feb 01 2021