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

7, 11, 17, 19, 29, 31, 47, 53, 59, 61, 67, 71, 83, 101, 113, 151, 157, 163, 167, 173, 181, 223, 227, 233, 239, 257, 269, 271, 277, 307, 313, 331, 337, 359, 373, 379, 383, 389, 431, 433, 463, 479, 487, 499, 521, 569, 587, 601, 619, 631, 641, 643, 653, 673, 677, 683, 691

Offset: 1

Author

N. J. A. Sloane, Dec 26 2017

Keywords

• nonn
• easy

Comments

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index to sequences related to decomposition of primes in quadratic fields.

Crossrefs

Cf. A105885, A296920, A296927, A296928.

Programs

Maple

Load the Maple program HH given in A296920. Then run HH(-13, 200); This produces A296926, A296927, A296928, A105885.

Mathematica

Select[Prime[Range[125]], KroneckerSymbol[-13, #] == 1 &] (* Amiram Eldar, Nov 17 2023 *)

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

Formula

a(n) ~ 2n log n. - Charles R Greathouse IV, Mar 18 2018

Primes == {1, 7, 9, 11, 15, 17, 19, 25, 29, 31, 47, 49} (mod 52). - Travis Scott, Jan 05 2023