A257552 Primes p such that q = p^2 - 2, r = q^2 - 2 and s = r^2 - 2 are also prime.

2, 3, 3299, 6323, 9127, 9697, 26357, 27061, 27809, 77513, 83299, 83641, 87701, 99721, 117307, 152123, 197969, 202987, 243461, 248179, 249397, 262121, 285721, 285823, 351217, 379273, 388009, 397763, 436477, 502063, 523777, 531263, 541661, 583501, 651881

Offset

1,1

Comments

Are there primes p > 2 such that t = s^2 - 2 is also prime?

t = s^2 - 2 is prime for p = 1644103, 3892831, 5178193, 5497949, 5657699, ... - Chai Wah Wu, Apr 30 2015

Links

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Example

3 is in the sequence because 3^2 - 2 = 7, 7^2 - 2 = 47 and 47^2 - 2 = 2207 are all primes.
5 is not in the sequence, because, although 5^2 - 2 = 23 is prime, 23^2 - 2 = 527 = 17 * 31.

Mathematica

Select[Prime@ Range@ 100000, PrimeQ[#^2 - 2] && PrimeQ[Nest[#^2 - 2 &, #, 2]] && PrimeQ[Nest[#^2 - 2 &, #, 3]] &] (* Michael De Vlieger, Apr 29 2015 *)
Select[Prime@Range@60000, PrimeQ[#^2 - 2] && PrimeQ[#^4 - 4 #^2 + 2] && PrimeQ[#^8 - 8 #^6 + 20 #^4 - 16 #^2 + 2] &] (* Vincenzo Librandi, Apr 30 2015 *)
Select[Prime[Range[10^4]], Union[PrimeQ[{#^2 - 2, #^4 - 4#^2 + 2, #^8 - 8#^6 + 20#^4 - 16#^2 + 2}]] == {True} &] (* Alonso del Arte, May 01 2015 *)

Prog

(Magma) [p: p in PrimesUpTo(1500000)| IsPrime(p^4-4*p^2+2)and IsPrime(p^2-2)and IsPrime(p^8-8*p^6+20*p^4-16*p^2+2)]; // Vincenzo Librandi, Apr 30 2015
(Python)
from gmpy2 import is_prime, next_prime
A257552_list, p = [], 2
for _ in range(10**9):
....q = p**2 - 2
....if is_prime(q):
........r = q**2 -2
........if is_prime(r):
............s = r**2-2
............if is_prime(s):
................A257552_list.append(p)
....p = next_prime(p) # Chai Wah Wu, Apr 30 2015
(Perl) use Math::GMP ":constant"; use ntheory ":all"; my($q, $r, $s); forprimes { say if is_prime($q=$_**2-2) && is_prime($r=$q**2-2) && is_prime($s=$r**2-2); } 1e9;  # Dana Jacobsen, May 02 2015

Crossrefs

Subsequence of A257551 and A062326.

Cf. A253264.

Sequence in context: A185156 A235935 A182383 * A038104 A290972 A097301

Adjacent sequences: A257549 A257550 A257551 * A257553 A257554 A257555

Keyword

nonn

Author

Zak Seidov, Apr 29 2015

Extensions

More terms from Vincenzo Librandi, Apr 30 2015

Status

approved