A122429 Primes p such that q = 4p^2 + 1, r = 4q^2 + 1 and s = 4r^2 + 1 are all primes.

13, 9833, 41647, 151607, 264757, 356123, 361223, 446863, 449093, 457813, 531383, 641057, 655927, 841697, 855947, 899263, 913687, 1052813, 1081757, 1379383, 1506493, 1575757, 1685087, 1821013, 1821377, 1981517, 2054233, 2142037

Offset: 1

Author

Zak Seidov, Oct 20 2006

Keywords

• nonn

Comments

Next terms up to 400000th prime are 2286877, 2524157, 2595247, 2621737, 2931583, 3023437, 3425843, 3428567, 3538517, 3705187, 3777883, 3799717, 3875143, 3913727, 3973553, 4019833, 4167073, 4249523, 4488167, 4651873, 4822193, 4914937, 5054167, 5108293, 5140147, 5465303, 5520007, 5542003. - Zak Seidov, Jan 16 2009

Subsequence of A122424. - Pierre CAMI, Jul 21 2014

Examples

			13 is there because 13, 677, 1833317 and 13444204889957 are prime.
		

References

• Clifford A. Pickover, A Passion for Mathematics, John Wiley & Sons, Inc., 2005, p.74.

Links

• Pierre CAMI, Table of n, a(n) for n = 1..3636

Crossrefs

Cf. A052291, A005574, A001912, A122424.

Programs

Mathematica

Reap[Do[p=Prime[n];q=4p^2+1;r=4q^2+1;s=4r^2+1;If[PrimeQ[{q,r,s}]=={True, True,True},Sow[p]],{n,15000}]][[2,1]]
Select[Prime[Range[200000]],AllTrue[NestList[4#^2+1&,#,3],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Nov 22 2015 *)

PARI

f(x)=4*x^2+1;
forprime(p=1, 10^8, if(isprime(f(p))&&isprime(f(f(p)))&&isprime(f(f(f(p)))), print1(p, ", "))) \\ Derek Orr, Jul 31 2014

Extensions

More terms from Don Reble, Oct 24 2006

Edited by R. J. Mathar, Nov 02 2009