cp's OEIS Frontend

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

%I A307250 #25 Aug 11 2025 08:13:09
%S A307250 193,233,257,277,349,389,397,461,509
%N A307250 Primes q == 1 (mod 4) such that there is exactly one equivalence class of quaternary quadratic forms of discriminant q not representing 2.
%C A307250 This list is exhaustive for q < 350000. - _Andy Huchala_, May 17 2023
%H A307250 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 A307250 Jürg Kramer, <a href="https://gdz.sub.uni-goettingen.de/id/PPN235181684_0281">On the linear independence of certain theta-series</a>, Mathematische Annalen 281.2 (1988): 219-228. See page 226.
%o A307250 (Sage)
%o A307250 bound = 100
%o A307250 P = Primes()
%o A307250 p = 3
%o A307250 for i in range(bound):
%o A307250     p = P.next(p)
%o A307250     if p % 4 == 1:
%o A307250         K1.<a> = NumberField(x^2 - p)
%o A307250         K2.<b> = NumberField(x^2 + p)
%o A307250         K3.<c> = NumberField(x^2 + 3*p)
%o A307250         zeta = K1.zeta_function()
%o A307250         h2 = len(K2.class_group())
%o A307250         h3 = len(K3.class_group())
%o A307250         H_plus = 1/2 * zeta(-1) + 1/8 * h2 + 1/6 * h3
%o A307250         H = (H_plus + int((p + 19)/24))/2
%o A307250         if abs(H_plus-H-1)<.01:
%o A307250              print(p) # _Andy Huchala_, May 17 2023
%K A307250 nonn,more
%O A307250 1,1
%A A307250 _N. J. A. Sloane_, Mar 31 2019
%E A307250 Name clarified by _Andy Huchala_, May 18 2023