A341229 Primes p such that (p^64 + 1)/2 is prime.

3, 353, 587, 727, 863, 883, 919, 1217, 1237, 1657, 2029, 2203, 2333, 3209, 3529, 3617, 3889, 4889, 5387, 5557, 5689, 5749, 6701, 6961, 7727, 8443, 9377, 9433, 10009, 10243, 10691, 10799, 11027, 12071, 12451, 13681, 13687, 15569, 15601, 15823, 16759, 17939

Offset: 1

Author

Jon E. Schoenfield, Feb 07 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, A340480, A341210, A341224, and this sequence list primes of the form (p^j + 1)/2 for j=2^0=1, j=2^1=2, ..., j=2^6=64, respectively.

Examples

			(3^64 + 1)/2 = 1716841910146256242328924544641 is prime, so 3 is a term.
(5^64 + 1)/2 = 271050543121376108501863200217485427856445313 = 769*3666499598977*96132956782643741951225664001, so 5 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), A340480 (k=3), A341210 (k=4), A341224 (k=5), (this sequence) (k=6).

Programs

Maple

q:= p-> (q-> q(p) and q((p^64+1)/2))(isprime):
select(q, [$3..20000])[];  # Alois P. Heinz, Feb 07 2021

Mathematica

Select[Range[18000], PrimeQ[#] && PrimeQ[(#^64 + 1)/2] &] (* Amiram Eldar, Feb 07 2021 *)

PARI

isok(p) = (p>2) && isprime(p) && ispseudoprime((p^64 + 1)/2); \\ Michel Marcus, Feb 07 2021