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A003630 Inert rational primes in Q(sqrt(3)).
(Formerly M3766)
5
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 (list; graph; refs; listen; history; text; internal format)
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

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

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.

Sequence in context: A099382 A163633 A092242 * A122565 A247607 A079016

Adjacent sequences:  A003627 A003628 A003629 * A003631 A003632 A003633

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Mira Bernstein

STATUS

approved

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Last modified November 18 19:06 EST 2017. Contains 294894 sequences.