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

%I #23 May 19 2023 08:03:57

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

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

%C This list is exhaustive for q < 350000. - _Andy Huchala_, May 17 2023

%H F. Hirzebruch, <a href="http://www.numdam.org/item/10.24033/asens.1342.pdf">Modulflächen und Modulkurven zur symmetrischen Hilbertschen Modulgruppe</a>, Annales scientifiques de l’É.N.S. 4e série, tome 11, no 1 (1978), p. 101-165. See page 135.

%H Jürg Kramer, <a href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN00233125X">On the linear independence of certain theta-series</a>, Mathematische Annalen 281.2 (1988): 219-228. See page 226.

%o (Sage)

%o bound = 100

%o P = Primes()

%o p = 3

%o for i in range(bound):

%o p = P.next(p)

%o if p % 4 == 1:

%o K1.<a> = NumberField(x^2 - p)

%o K2.<b> = NumberField(x^2 + p)

%o K3.<c> = NumberField(x^2 + 3*p)

%o zeta = K1.zeta_function()

%o h2 = len(K2.class_group())

%o h3 = len(K3.class_group())

%o H_plus = 1/2 * zeta(-1) + 1/8 * h2 + 1/6 * h3

%o H = (H_plus + int((p + 19)/24))/2

%o if abs(H_plus-H-1)<.01:

%o print(p) # _Andy Huchala_, May 17 2023

%K nonn,more

%O 1,1

%A _N. J. A. Sloane_, Mar 31 2019

%E Name clarified by _Andy Huchala_, May 18 2023