A363147 Primes q == 1 (mod 4) such that there is at least one equivalence class of quaternary quadratic forms of discriminant q not representing 2.

193, 233, 241, 257, 277, 281, 313, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 541, 557, 569, 577, 593, 601, 613, 617, 641, 653, 661, 673, 677, 701, 709, 733, 757, 761, 769, 773, 797, 809, 821, 829, 853, 857, 877, 881, 929, 937

Offset

1,1

Links

Andy Huchala, Table of n, a(n) for n = 1..20000

F. Hirzebruch, Modulflächen und Modulkurven zur symmetrischen Hilbertschen Modulgruppe, Annales scientifiques de l’É.N.S. 4e série, tome 11, no 1 (1978), p. 101-165. See page 135.

Jürg Kramer, On the linear independence of certain theta-series, Mathematische Annalen 281.2 (1988): 219-228. See page 226.

Prog

(Sage)
bound = 100
P = Primes()
p = 2
for i in range(bound):
    p = P.next(p)
    if p % 4 == 1:
        K1.<a> = NumberField(x^2 - p)
        K2.<b> = NumberField(x^2 + p)
        K3.<c> = NumberField(x^2 + 3*p)
        zeta = K1.zeta_function()
        h2 = len(K2.class_group())
        h3 = len(K3.class_group())
        H_plus = int(abs(.49+1/2*zeta(-1)+1/8 * h2 + 1/6*h3))
        H = (H_plus+int((p + 19)/24))/2
        if H_plus-H>0:
            print(p)

Crossrefs

Cf. A307250, A363148.

Sequence in context: A298728 A332457 A088119 * A307250 A226147 A142925

Adjacent sequences: A363144 A363145 A363146 * A363148 A363149 A363150

Keyword

nonn

Author

Andy Huchala, May 18 2023

Status

approved