A263849 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 nu in R of norm m = A031363(n) can be written as a sum of three squares in R in at least k ways.

1, 6, 12, 24, 32, 24, 54, 24, 24, 30, 24, 48, 48, 96, 24, 48, 96, 48, 24, 120, 108, 48, 72, 48, 120, 54, 48, 48, 48, 84, 72, 120, 72, 78, 48, 144, 72, 72, 128, 192, 120, 96, 48, 48, 96, 96, 216, 72, 48, 120, 96, 96, 48, 96, 48, 120, 96, 224, 72, 120, 48, 288, 72, 48, 72, 246, 240, 120, 144

Offset: 0

Author

N. J. A. Sloane, Nov 15 2015

Keywords

• nonn

Comments

Let R = Z[(1+sqrt{5})/2] denote the ring of integers in the real quadratic number field of discriminant 5. The main result of Maass (1941) is that every totally positive nu in R is a sum of 3 squares x^2+y^2+z^2 with x,y,z in R. The number N_{nu} of such representations is given by the formula in the theorem on page 191. The norms of the totally positive elements nu are rational integers m belonging to A031363, so we can order the terms of the sequence according to the values m = A031363(n). [Comment based on remarks from Gabriele Nebe.]

The terms were computed with the aid of Magma by David Durstoff, Nov 11 2015.

The attached file from David Durstoff gives list of pairs m=A031363(n), a(n), and also the initial terms of Maass's series theta(tau). David Durstoff says: "I expressed theta(tau) in terms of two variables q1 and q2. The coefficient of q1^k q2^m is a(nu) with k = trace(nu/delta) and m = trace(nu), where delta = (5+sqrt{5})/2 is a generator of the different ideal. I computed the terms for q1^0 to q1^10 and all possible powers of q2."

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. E.g. there are 78 ways of writing (27 + 7*sqrt(5))/2 as a sum of three squares in R, and there are 192 ways of writing 11 as a sum of three squares in R, yet both elements have norm 121. The sequence of possible norms for which this can occur is 121, 209, 341, 361, 451, 484, 551, 589, 605, 649, ... - Robin Visser, Mar 28 2025

Examples

			From _Robin Visser_, Mar 30 2025: (Start)
a(1) = 6, as every totally positive element of norm A031363(1)=1 in R can be written as a sum of three squares in R in exactly 6 ways.  E.g. the element 1 in R has norm 1 and can be written as a sum of three squares in R as:
  1 = 1^2 + 0^2 + 0^2 = (-1)^2 + 0^2 + 0^2 = 0^2 + 1^2 + 0^2 = 0^2 + (-1)^2 + 0^2 = 0^2 + 0^2 + 1^2 = 0^2 + 0^2 + (-1)^2.
a(2) = 12, as every totally positive element of norm A031363(2)=4 in R can be written as a sum of three squares in R in exactly 12 ways. E.g. the element 2 in R has norm 4 and can be written as a sum of three squares in R as:
  2 = 1^2 + 1^2 + 0^2 = 1^2 + 0^2 + 1^2 = 0^2 + 1^2 + 1^2 = 1^2 + (-1)^2 + 0^2 = 1^2 + 0^2 + (-1)^2 = 0^2 + 1^2 + (-1)^2 = (-1)^2 + 1^2 + 0^2 = (-1)^2 + 0^2 + 1^2 = 0^2 + (-1)^2 + 1^2 = (-1)^2 + (-1)^2 + 0^2 = (-1)^2 + 0^2 + (-1)^2 = 0^2 + (-1)^2 + (-1)^2.
a(3) = 24, as every totally positive element of norm A031363(3)=5 in R can be written as a sum of three squares in R in exactly 24 ways. E.g. the element (5+sqrt(5))/2 in R has norm 5 and can be written as a sum of three squares in R as:
  (5+sqrt(5))/2 = w^2 + 1^2 + 0^2 = w^2 + 0^2 + 1^2 = 0^2 + w^2 + 1^2 = 1^2 + w^2 + 0^2 = 0^2 + 1^2 + w^2 = 1^2 + 0^2 + w^2 = (-w)^2 + 1^2 + 0^2 = (-w)^2 + 0^2 + 1^2 = 0^2 + (-w)^2 + 1^2 = 1^2 + (-w)^2 + 0^2 = 0^2 + 1^2 + (-w)^2 = 1^2 + 0^2 + (-w)^2 = w^2 + (-1)^2 + 0^2 = w^2 + 0^2 + (-1)^2 = 0^2 + w^2 + (-1)^2 = (-1)^2 + w^2 + 0^2 = 0^2 + (-1)^2 + w^2 = (-1)^2 + 0^2 + w^2 = (-w)^2 + (-1)^2 + 0^2 = (-w)^2 + 0^2 + (-1)^2 = 0^2 + (-w)^2 + (-1)^2 = (-1)^2 + (-w)^2 + 0^2 = 0^2 + (-1)^2 + (-w)^2 = (-1)^2 + 0^2 + (-w)^2, where w = (1+sqrt(5))/2. (End)
		

References

• 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.

Links

• Robin Visser, Table of n, a(n) for n = 0..6000
• Harvey Cohn, A computation of some bi-quadratic class numbers, Math. Tables Aids Comput. 12 (1958), 213-217. See pages 215-217.
• David Durstoff, Table showing list of pairs m=A031363(n), a(n)

Crossrefs

Cf. A031363 (the norms), A035187 (number of ideals with that norm).

See A263850 for another version of this sequence.

Cf. A005875 (sum of 3 squares in Z), A000118 (sum of 4 squares in Z).

Extensions

More terms from Robin Visser, Mar 28 2025