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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.

%I A363147 #16 Aug 11 2025 08:53:43
%S A363147 193,233,241,257,277,281,313,337,349,353,373,389,397,401,409,421,433,
%T A363147 449,457,461,509,521,541,557,569,577,593,601,613,617,641,653,661,673,
%U A363147 677,701,709,733,757,761,769,773,797,809,821,829,853,857,877,881,929,937
%N 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.
%H A363147 Andy Huchala, <a href="/A363147/b363147.txt">Table of n, a(n) for n = 1..20000</a>
%H A363147 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 A363147 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 A363147 (Sage)
%o A363147 bound = 100
%o A363147 P = Primes()
%o A363147 p = 2
%o A363147 for i in range(bound):
%o A363147     p = P.next(p)
%o A363147     if p % 4 == 1:
%o A363147         K1.<a> = NumberField(x^2 - p)
%o A363147         K2.<b> = NumberField(x^2 + p)
%o A363147         K3.<c> = NumberField(x^2 + 3*p)
%o A363147         zeta = K1.zeta_function()
%o A363147         h2 = len(K2.class_group())
%o A363147         h3 = len(K3.class_group())
%o A363147         H_plus = int(abs(.49+1/2*zeta(-1)+1/8 * h2 + 1/6*h3))
%o A363147         H = (H_plus+int((p + 19)/24))/2
%o A363147         if H_plus-H>0:
%o A363147             print(p)
%Y A363147 Cf. A307250, A363148.
%K A363147 nonn
%O A363147 1,1
%A A363147 _Andy Huchala_, May 18 2023