A383067 The set of positive integers k which can be expressed as a sum of two units in some cyclic cubic field.

1, 2, 3, 4, 5, 7, 19, 22

Offset: 1

Author

Robin Visser, Apr 15 2025

Keywords

• nonn
• fini
• full

Comments

These are all the positive integers k such that there exists some cubic number field K whose Galois group is cyclic (C_3) and contains units u, v in K such that k = u + v.

Examples

			For each positive integer k given in the sequence, it can be written as a sum of two units in some cyclic cubic field as follows:
1 = u + (-u+1), where u is a root of x^3 + x^2 - 2x - 1.
2 = u + (-u+2), where u is a root of x^3 - 3x - 1.
3 = (u^2) + (-u^2+3), where u is a root of x^3 + x^2 - 2x - 1.
4 = (u^2+2u) + (-u^2-2u+4), where u is a root of x^3 + x^2 - 2x - 1.
5 = (u^2-u) + (-u^2+u+5), where u is a root of x^3 + x^2 - 2x - 1.
7 = (u^2) + (-u^2+7), where u is a root of x^3 - x^2 - 4x - 1.
19 = (5u^2+9u) + (-5u^2-9u+19), where u is a root of x^3 + x^2 - 2x - 1.
22 = (4u^2-5u) + (-4u^2+5u+22), where u is a root of x^3 + x^2 - 2x - 1.
		

Links

• M. Tinková, R. Visser, and P. Yatsyna, Sums of two units in number fields, arXiv:2502.01345 [math.NT], 2025.
• I. Vukusic and V. Ziegler, On a family of unit equations over simplest cubic fields, J. Théor. Nombres Bordeaux 34 (2022), no. 3, 705-718.

Crossrefs

Cf. A383068.