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A307250
Primes q == 1 (mod 4) such that there is exactly one equivalence class of quaternary quadratic forms of discriminant q not representing 2.
2
193, 233, 257, 277, 349, 389, 397, 461, 509
OFFSET
1,1
COMMENTS
This list is exhaustive for q < 350000. - Andy Huchala, May 17 2023
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 = 3
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 = 1/2 * zeta(-1) + 1/8 * h2 + 1/6 * h3
H = (H_plus + int((p + 19)/24))/2
if abs(H_plus-H-1)<.01:
print(p) # Andy Huchala, May 17 2023
CROSSREFS
Sequence in context: A332457 A088119 A363147 * A226147 A142925 A060333
KEYWORD
nonn,more
AUTHOR
N. J. A. Sloane, Mar 31 2019
EXTENSIONS
Name clarified by Andy Huchala, May 18 2023
STATUS
approved