A245674 Prime numbers P such that 8*P^2-1 and 8*(8*P^2-1)^2-1 are also prime numbers.

2, 79, 107, 173, 257, 359, 383, 523, 593, 971, 1493, 1811, 1867, 2273, 2357, 3187, 4111, 4723, 6389, 7607, 8101, 8699, 9473, 11027, 12157, 12227, 15017, 16301, 16987, 18797, 19801, 19913, 20071, 20323, 21313, 22003, 22307, 23203, 24229, 24733, 24859, 24943

Offset: 1

Author

Pierre CAMI, Jul 29 2014

Keywords

• nonn
• easy

Comments

Subsequence of A245639.

For P < 10^9 in this sequence, 8*(8*(8*P^2-1)^2-1)^2-1 is composite.

Let f(x) = 8*x^2-1 and P >= 2. Then {f(P), f(f(P)), f(f(f(P)))} cannot all be prime. Proof: By doing cases on P mod 7, it can be shown that {f(P), f(f(P)), f(f(f(P)))} contains a multiple of 7. Also, all 3 numbers are greater than 7. - Jason Yuen, Feb 26 2025

Examples

			2 is prime, 8*2^2-1=31 is prime, 8*31^2-1=7687 is prime, so 2 is in the sequence.
		

Links

• Pierre CAMI, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A245639.

Programs

Magma

[p: p in PrimesUpTo(25000)| IsPrime(8*p^2-1)and IsPrime(512*p^4-128*p^2+7)]; // Vincenzo Librandi, Sep 08 2014

Mathematica

f[n_]:=8 n^2 - 1; Select[Prime[Range[3000]], PrimeQ[f[#]]&&PrimeQ[f[f[#]]]&] (* Vincenzo Librandi, Sep 08 2014 *)

PARI

f(x) = 8*x^2-1;
forprime(p=1,10^5,if(ispseudoprime(f(p)) && ispseudoprime(f(f(p))), print1(p,", "))) \\ Derek Orr, Jul 29 2014