A003630 Inert rational primes in Q[sqrt(3)]. (Formerly M3766)

5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, 101, 103, 113, 127, 137, 139, 149, 151, 163, 173, 197, 199, 211, 223, 233, 257, 269, 271, 281, 283, 293, 307, 317, 331, 353, 367, 379, 389, 401, 439, 449, 461, 463, 487, 499, 509, 521, 523, 547, 557, 569, 571, 593

Offset

1,1

Comments

Primes p such that p divides 3^(p-1)/2 + 1. - Cino Hilliard, Sep 04 2004

Primes p such that 1 + 4*x + x^2 is irreducible over GF(p). - Joerg Arndt, Aug 10 2011

Conjecture: Primes congruent to {5, 7} mod 12. - Vincenzo Librandi, Aug 06 2012

The above conjecture is correct. In fact, this is the sequence of primes p such that Kronecker(12,p) = -1 (12 is the discriminant of Q[sqrt(3)]), that is, odd primes that have 3 as a quadratic nonresidue. - Jianing Song, Nov 21 2018

Conjecture: Let r(n) = (a(n) - 1)/(a(n) + 1) if a(n) mod 4 = 1, (a(n) + 1)/(a(n) - 1) otherwise; then Product_{n>=1} r(n) = (2/3) * (4/3) * (8/9) * (10/9) * (14/15) * ... = sqrt(3)/2. (See A010527.) We see that the sum of the numerator and denominator of each fraction equals the corresponding term of the sequence: 2 + 3 = 5, 4 + 3 = 7, 8 + 9 = 17, ... - Dimitris Valianatos, Mar 26 2017

References

H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

T. D. Noe, Table of n, a(n) for n = 1..1000

Index to sequences related to decomposition of primes in quadratic fields

Example

Since (-1)*(1 - sqrt(3))*(1 + sqrt(3)) = 2, 2 is not in the sequence.
3 is not in the sequence for obvious reasons.
x^2 == 3 (mod 5) has no solution, which means that 5 is an inert prime in Z[sqrt(3)]. Therefore, 5 is in the sequence.

Mathematica

Select[Prime[Range[2, 200]], JacobiSymbol[3, #] == -1 &] (* Alonso del Arte, Mar 26 2017 *)

Prog

(PARI) {a(n) = local( cnt, m ); if( n<1, return( 0 )); while( cnt < n, if( isprime( m++) && kronecker( 12, m )== -1, cnt++ )); m} /* Michael Somos, Aug 14 2012 */

Crossrefs

Cf. A010527, A097933.

Sequence in context: A099382 A163633 A092242 * A122565 A369105 A247607

Adjacent sequences: A003627 A003628 A003629 * A003631 A003632 A003633

Keyword

nonn,easy

Author

N. J. A. Sloane, Mira Bernstein

Status

approved