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%I M3766 #76 May 24 2022 02:44:42
%S 5,7,17,19,29,31,41,43,53,67,79,89,101,103,113,127,137,139,149,151,
%T 163,173,197,199,211,223,233,257,269,271,281,283,293,307,317,331,353,
%U 367,379,389,401,439,449,461,463,487,499,509,521,523,547,557,569,571,593
%N Inert rational primes in Q[sqrt(3)].
%C Primes p such that p divides 3^(p-1)/2 + 1. - _Cino Hilliard_, Sep 04 2004
%C Primes p such that 1 + 4*x + x^2 is irreducible over GF(p). - _Joerg Arndt_, Aug 10 2011
%C Conjecture: Primes congruent to {5, 7} mod 12. - _Vincenzo Librandi_, Aug 06 2012
%C 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
%C 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
%D H. Hasse, Number Theory, Springer-Verlag, NY, 1980, p. 498.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A003630/b003630.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>
%e Since (-1)*(1 - sqrt(3))*(1 + sqrt(3)) = 2, 2 is not in the sequence.
%e 3 is not in the sequence for obvious reasons.
%e 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.
%t Select[Prime[Range[2, 200]], JacobiSymbol[3, #] == -1 &] (* _Alonso del Arte_, Mar 26 2017 *)
%o (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 */
%Y Cf. A010527, A097933.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, _Mira Bernstein_
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