OFFSET
1,1
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
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
MAPLE
MATHEMATICA
Select[Prime[Range[125]], KroneckerSymbol[-13, #] == 1 &] (* Amiram Eldar, Nov 17 2023 *)
PROG
(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
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Dec 26 2017
STATUS
approved