A307250 Primes q == 1 (mod 4) such that there is exactly one equivalence class of quaternary quadratic forms of discriminant q not representing 2.

193, 233, 257, 277, 349, 389, 397, 461, 509

Offset: 1

Author

N. J. A. Sloane, Mar 31 2019

Keywords

• nonn
• more

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.

Programs

Sage

bound = 100
P = Primes()
p = 3
for i in range(bound):
    p = P.next(p)
    if p % 4 == 1:
        K1. = NumberField(x^2 - p)
        K2. = NumberField(x^2 + p)
        K3. = 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

Extensions

Name clarified by Andy Huchala, May 18 2023