A225431 Primes p such that there is a prime q satisfying 3*p^2 - q^2 = 2.

3, 11, 41, 2131, 110771, 15558008491

Offset: 1

Author

Irina Gerasimova, May 07 2013

Keywords

• nonn

Comments

Primes q: 5, 19, 71, 3691, 191861,...

(q - p)/2: 1, 4, 15, 780, 40545,...

a(7) > 2.8016852867294*10^4857. - Zak Seidov, May 09 2013

Probably finite.

This is a form of Pell's equation with the requirement that solutions be prime. - T. D. Noe, May 14 2013

Examples

			11 is prime and sqrt(3*11^2 - 2) = sqrt(361) = 19 is prime, so 11 is in the sequence.
		

Links

• Eric W. Weisstein, MathWorld: Pell Equation

Programs

Mathematica

nn = 1000; ta = LinearRecurrence[{4, -1}, {1, 3}, nn]; tb = LinearRecurrence[{4, -1}, {1, 5}, nn]; sol = Select[Range[nn], PrimeQ[ta[[#]]] && PrimeQ[tb[[#]]] &]; ta[[sol]] (* T. D. Noe, May 14 2013 *)

PARI

v=[1,1]; for(i=1,1e4,v=[v[2],4*v[2]-v[1]]; if(ispseudoprime(v[2]) && ispseudoprime(sqrtint(3*v[2]^2-2)), print1(v[2]", "))) \\ Charles R Greathouse IV, May 13 2013

PFGW

ABC2 Linear(3,11,41,153,$a) & Linear(5,19,71,265,$a)
a: from 3 to 20000 // Charles R Greathouse IV, May 13 2013

Extensions

a(4) from R. J. Mathar, May 07 2013

a(6) from Charles R Greathouse IV, May 07 2013

a(5) from Zak Seidov, May 09 2013