A387575 The maximal norm of an additively indecomposable element in Shanks' simplest cubic field Q[x]/(x^3 - n*x^2 - (n+3)*x - 1).

9, 13, 19, 9, 47, 7, 147, 223, 337, 507, 701, 965, 7, 1721, 2231, 2883, 3595, 4471, 5547, 6707, 8093, 855, 11513, 13577, 15987, 18541, 21475, 24843, 28391, 32411, 2331, 41735, 47081, 53067, 59317, 66253, 73947, 81953, 90767, 183, 110531, 1525, 133563, 146011, 159541, 174243, 189425, 205841

Offset

0,1

Comments

For any totally real field K, an additively indecomposable element of K is a totally positive element in the maximal order of K which cannot be written as the sum of two totally positive integral elements of K. Here, an element x of K is totally positive if all conjugates of x are positive real numbers.

Let K be a simplest cubic field with defining polynomial f(x) = x^3 - n*x^2 - (n+3)*x - 1, and let rho be a root of f(x). Kala and Tinková classified all additively indecomposable elements of K in the case where the ring of integers O_K is Z[rho], and Gil-Muñoz and Tinková classified all additively indecomposable elements of K in the case where Z[rho] has index 3 in O_K.

Links

Robin Visser, Table of n, a(n) for n = 0..240

Daniel Gil-Muñoz and Magdaléna Tinková, Additive structure of non-monogenic simplest cubic fields, Ramanujan J. 66 (2025), no. 3, Paper No. 47, 56 pp.

Vítězslav Kala, Universal quadratic forms and indecomposables in number fields: a survey, Commun. Math. 31 (2023), no. 2, 81-114.

Vítězslav Kala and Magdaléna Tinková, Universal quadratic forms, small norms, and traces in families of number fields, Int. Math. Res. Not. IMRN 2023, no. 9, 7541-7577.

Daniel Shanks, The simplest cubic fields, Math. Comp., 28 (1974), 1137-1152.

Magdaléna Tinková, Trace and norm of indecomposable integers in cubic orders, Ramanujan J. 61 (2023), no. 4, 1121-1144.

Formula

If n > 3 and the maximal order of Q[x]/(x^3 - n*x^2 - (n+3)*x - 1) is generated by a root of x^3 - n*x^2 - (n+3)*x - 1, then a(n) = ((n^2 + 3*n + 9)^2)/27 if n == 0 (mod 3), else a(n) = (n^4 + 6*n^3 + 24*n^2 + 47*n + 57)/27 if n == 1 (mod 3), else a(n) == (n^4 + 6*n^3 + 24*n^2 + 43*n + 51)/27 if n = 2 (mod 3) [Tinková, Theorem 1.2].

If the order Z[rho] generated by a root rho of x^3 - n*x^2 - (n+3)*x - 1 has index 3 in the maximal order of Q[x]/(x^3 - n*x^2 - (n+3)*x - 1), then a(n) = (2*n^3 + 9*n^2 + 27*n + 27)/27 if n <= 48, otherwise a(n) = ((n^2 + 3*n + 9)^2)/729 [Gil-Muñoz--Tinková, Proposition 6.4]. - Robin Visser, Nov 07 2025

Example

For n = 0, every additively indecomposable element in Q[x]/(x^3 - 3*x - 1) has norm either 1, 3, or 9, thus a(0) = 9.
For n = 1, every additively indecomposable element in Q[x]/(x^3 - x^2 - 4*x - 1) has norm either 1, 5, or 13, thus a(1) = 13.
For n = 2, every additively indecomposable element in Q[x]/(x^3 - 2*x^2 - 5*x - 1) has norm either 1, 7, 11, or 19, thus a(2) = 19.

Crossrefs

Cf. A005472, A387207, A387574.

Sequence in context: A263941 A090415 A026283 * A307287 A373781 A373914

Adjacent sequences: A387572 A387573 A387574 * A387576 A387577 A387578

Keyword

nonn

Author

Robin Visser, Sep 02 2025

Status

approved