cp's OEIS Frontend

A263850 Let R = Z[(1+sqrt(5))/2] denote the ring of integers in the real quadratic number field of discriminant 5. Then a(n) is the largest integer k such that every totally positive element of norm n in R can be written as a sum of three squares in R in at least k ways, or 0 if there is no totally positive element of norm n.

%I A263850 #19 Apr 06 2025 15:02:46
%S A263850 1,6,0,0,12,24,0,0,0,32,0,24,0,0,0,0,54,0,0,24,24,0,0,0,0,30,0,0,0,24,
%T A263850 0,48,0,0,0,0,48,0,0,0,0,96,0,0,24,48,0,0,0,96,0,0,0,0,0,48,0,0,0,24,
%U A263850 0,120,0,0,108,0,0,0,0,0,0,48,0,0,0,0,72,0,0,48,120,54,0,0,0,0,0,0,0,48,0,0,0
%N A263850 Let R = Z[(1+sqrt(5))/2] denote the ring of integers in the real quadratic number field of discriminant 5. Then a(n) is the largest integer k such that every totally positive element of norm n in R can be written as a sum of three squares in R in at least k ways, or 0 if there is no totally positive element of norm n.
%C A263850 The terms were computed with the aid of Magma by David Durstoff, Nov 11 2015. See A263849 for further information.
%C A263850 Note that there are examples of totally positive elements x and y in R which have the same norm, but for which the number of ways of writing x as a sum of three squares in R is different to the number of ways of writing y as a sum of three squares in R. See A263849 for explicit examples. - _Robin Visser_, Mar 28 2025
%D A263850 Maass, Hans. Über die Darstellung total positiver Zahlen des Körpers R (sqrt(5)) als Summe von drei Quadraten, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. Vol. 14. No. 1, pp. 185-191, 1941.
%H A263850 Robin Visser, <a href="/A263850/b263850.txt">Table of n, a(n) for n = 0..10000</a>
%H A263850 Harvey Cohn, <a href="https://doi.org/10.2307/2002024">A computation of some bi-quadratic class numbers</a>, Math. Tables Aids Comput. 12 (1958), 213-217.  See pages 215-217.
%o A263850 (Sage)
%o A263850 import itertools
%o A263850 def a(n):
%o A263850     if n==0: return 1
%o A263850     if any([((r[0]%5 in [2,3]) and (r[1]%2==1)) for r in factor(n)]): return 0
%o A263850     K.<w> = NumberField(x^2-x-1); cw = w.coordinates_in_terms_of_powers(); ans = []
%o A263850     for idl in K.ideals_of_bdd_norm(n)[n]:
%o A263850         for u in [1,-1,w,-w]:
%o A263850             X,Y = cw(u*idl.gens_reduced()[0]); num = 0
%o A263850             if (X < 0): continue
%o A263850             for b in range(-isqrt(X), isqrt(X)+1):
%o A263850                 for d in range(-isqrt(X-b^2), isqrt(X-b^2)+1):
%o A263850                     for f in range(-isqrt(X-b^2-d^2), isqrt(X-b^2-d^2)+1):
%o A263850                         S = b^2+d^2+f^2; M = isqrt(X-S);
%o A263850                         for (a,c,e) in itertools.product(range(-M, M+1), repeat=3):
%o A263850                             if (a^2+c^2+e^2+S==X) and (2*(a*b+c*d+e*f)+S==Y): num += 1
%o A263850             if (num > 0): ans.append(num)
%o A263850     return min(ans)  # _Robin Visser_, Mar 28 2025
%Y A263850 Cf. A263849 (another version of this sequence), A031363, A035187.
%Y A263850 Cf. A005875 (sum of 3 squares in Z), A000118 (sum of 4 squares in Z).
%K A263850 nonn
%O A263850 0,2
%A A263850 _N. J. A. Sloane_, Nov 18 2015