cp's OEIS Frontend

A296926 Rational primes that decompose in the field Q(sqrt(-13)).

%I A296926 #40 Nov 17 2023 07:34:12
%S A296926 7,11,17,19,29,31,47,53,59,61,67,71,83,101,113,151,157,163,167,173,
%T A296926 181,223,227,233,239,257,269,271,277,307,313,331,337,359,373,379,383,
%U A296926 389,431,433,463,479,487,499,521,569,587,601,619,631,641,643,653,673,677,683,691
%N A296926 Rational primes that decompose in the field Q(sqrt(-13)).
%C A296926 In general, primes that decompose in Q(sqrt(-p prime)) are congruent modulo 4p to t(-1)^[t^(phi(p)/2) mod p = 1 XOR t mod min(e,4) = 1], where t are the totatives of 2p, e is the even part of phi(p), and [P] returns 1 if P else 0. In other words, if phi(p) is at least twice even, then the t are signed so that the quadratic residuosity of t mod p aligns with the congruence of +-t mod 4 to 1--the modulus 4p is thence irreducible--; if only once, then the signature simply indicates quadratic residues modulo p. The imbalance of signs in either flank (t < p, t > p) of the signature also gives the class number of Q(sqrt(-p)), up to an excess factor of 3 if p == 3 (mod 8) but != 3. [E.g., for p = 13 we have +--+++ or +++--+, so the class number of Q(sqrt(-13)) = 2; for p = 11 == 3 (mod 8) we have +++-+ or -+---, so the class number of Q(sqrt(-11)) = 3/3 = 1.] - _Travis Scott_, Jan 05 2023
%H A296926 Amiram Eldar, <a href="/A296926/b296926.txt">Table of n, a(n) for n = 1..10000</a>
%H A296926 <a href="/index/Pri#primes_decomp_of">Index to sequences related to decomposition of primes in quadratic fields</a>.
%F A296926 a(n) ~ 2n log n. - _Charles R Greathouse IV_, Mar 18 2018
%F A296926 Primes == {1, 7, 9, 11, 15, 17, 19, 25, 29, 31, 47, 49} (mod 52). - _Travis Scott_, Jan 05 2023
%p A296926 Load the Maple program HH given in A296920. Then run HH(-13, 200); This produces A296926, A296927, A296928, A105885.
%t A296926 Select[Prime[Range[125]], KroneckerSymbol[-13, #] == 1 &] (* _Amiram Eldar_, Nov 17 2023 *)
%o A296926 (PARI) list(lim)=my(v=List()); forprime(p=5,lim, if(kronecker(-13,p)==1, listput(v,p))); Vec(v) \\ _Charles R Greathouse IV_, Mar 18 2018
%Y A296926 Cf. A105885, A296920, A296927, A296928.
%K A296926 nonn,easy
%O A296926 1,1
%A A296926 _N. J. A. Sloane_, Dec 26 2017