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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.
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
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
Sequence in context: A298728 A332457 A088119 * A307250 A226147 A142925
KEYWORD
nonn
AUTHOR
Andy Huchala, May 18 2023
STATUS
approved