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A225431
Primes p such that there is a prime q satisfying 3*p^2 - q^2 = 2.
1
3, 11, 41, 2131, 110771, 15558008491
OFFSET
1,1
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
EXAMPLE
11 is prime and sqrt(3*11^2 - 2) = sqrt(361) = 19 is prime, so 11 is in the sequence.
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 *)
PROG
(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
CROSSREFS
Sequence in context: A102417 A358426 A096147 * A359248 A099489 A077830
KEYWORD
nonn
AUTHOR
Irina Gerasimova, May 07 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
STATUS
approved