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A097933 Primes such that p divides 3^((p-1)/2) - 1. 9

%I

%S 11,13,23,37,47,59,61,71,73,83,97,107,109,131,157,167,179,181,191,193,

%T 227,229,239,241,251,263,277,311,313,337,347,349,359,373,383,397,409,

%U 419,421,431,433,443,457,467,479,491,503,541,563,577,587,599,601,613

%N Primes such that p divides 3^((p-1)/2) - 1.

%C Rational primes that decompose in the field Q[sqrt(3)]. - _N. J. A. Sloane_, Dec 26 2017

%C For all primes p > 2 and integers gcd(x, y, p) = 1, x^((p-1)/2) +- y^((p-1)/2) is divisible by p. This is because (x^((p-1)/2) - y^((p-1)/2))(x^((p-1)/2) + y^((p-1)/2)) = x^(p-1) - y^(p-1) is divisible by p according to Fermat's Little Theorem (FLT). This sequence lists p that divides 3^((p-1)/2) - 1^((p-1)/2), and A003630 lists the '+' case.

%C Apart from initial terms, this and A038874 are the same. - _N. J. A. Sloane_, May 31 2009

%C Primes in A091998. - _Reinhard Zumkeller_, Jan 07 2012

%C Also, primes congruent to 1 or 11 (mod 12). - _Vincenzo Librandi_, Mar 23 2013

%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) = (6/5) * (6/7) * (12/11) * (18/19) * ... = 2/sqrt(3). - _Dimitris Valianatos_, Mar 27 2017

%C 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 residue. - _Jianing Song_, Nov 21 2018

%H Reinhard Zumkeller, <a href="/A097933/b097933.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="https://oeis.org/wiki/Index_to_OEIS:_Section_Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>

%e For p = 5, 3^2 - 1 = 8 <> 3*k for any integer k, so 5 is not in this sequence.

%e For p = 11, 3^5 - 1 = 242 = 11*22, so 11 is in this sequence.

%t Select[Prime[Range[300]], MemberQ[{1, 11, 13, 23}, Mod[#, 24]]&] (* _Vincenzo Librandi_, Mar 23 2013 *)

%o (PARI) /* s = +-1, d=diff */ ptopm1d2(n,x,d,s) = { forprime(p=3,n,p2=(p-1)/2; y=x^p2 + s*(x-d)^p2; if(y%p==0,print1(p","))) }

%o (PARI) {a(n)= local(m, c); if(n<1, 0, c=0; m=0; while( c<n, m++; if( isprime(m)& kronecker(3,m)==1, c++)); m)} /* _Michael Somos_, Aug 28 2006 */

%o (Haskell)

%o a097933 n = a097933_list !! (n-1)

%o a097933_list = [x | x <- a091998_list, a010051 x == 1]

%o -- _Reinhard Zumkeller_, Jan 07 2012

%o (MAGMA) [p: p in PrimesUpTo(1000) | p mod 24 in [1, 11, 13, 23]]; // _Vincenzo Librandi_, Mar 23 2013

%Y Cf. A003630, A010051, A038874, A091998.

%K nonn

%O 1,1

%A _Cino Hilliard_, Sep 04 2004

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Last modified January 26 08:18 EST 2020. Contains 331278 sequences. (Running on oeis4.)