Definition
A Löschian number is a non‑negative integer that can be expressed in the form
$$ n = x^{2}+xy+y^{2}, $$
where $x$ and $y$ are integers. Equivalently, a Löschian number is the norm of an element of the ring of Eisenstein integers $\mathbb{Z}[\omega]$ (with $\omega = e^{2\pi i/3}$).
Overview
The set of Löschian numbers arises naturally in the study of quadratic forms, algebraic number theory, and lattice geometry. Because the expression $x^{2}+xy+y^{2}$ is invariant under the action of the hexagonal (or triangular) lattice, Löschian numbers enumerate distances squared between points in that lattice. They also correspond to norms in the quadratic integer domain $\mathbb{Z}[\omega]$, making them relevant to factorisation properties and class‑number calculations within that domain.
Etymology / Origin
The term is named after the German mathematician Wilhelm Lösch (1867–1935), who investigated representations of integers by quadratic forms of discriminant $-3$. The designation “Löschian” reflects his contributions to the classification of numbers represented by the form $x^{2}+xy+y^{2}$.
Characteristics
| Property | Description |
|---|---|
| Norm representation | For any Eisenstein integer $a + b\omega$, the norm is $N(a+b\omega)=a^{2}-ab+b^{2}$. By substituting $(x,y) = (a,-b)$ the norm takes the form $x^{2}+xy+y^{2}$; thus every Löschian number is a norm in $\mathbb{Z}[\omega]$. |
| Parity | A Löschian number is congruent to $0$ or $1$ modulo $3$. No integer congruent to $2 \pmod{3}$ can be expressed as $x^{2}+xy+y^{2}$. |
| Prime characterization | A rational prime $p$ is a Löschian number (i.e., can be written as $x^{2}+xy+y^{2}$ with $(x,y) |
| eq (0,0)$) if and only if $p = 3$ or $p \equiv 1 \pmod{3}$. Primes congruent to $2 \pmod{3}$ are not Löschian. | |
| Closure under multiplication | The product of two Löschian numbers is again a Löschian number, reflecting the multiplicative property of norms in $\mathbb{Z}[\omega]$. |
| Density | The asymptotic density of Löschian numbers among the non‑negative integers is $0$; however, their counting function grows on the order of $C,x$ with constant $C = \frac{2}{\sqrt{3},\pi}$. |
| Geometric interpretation | In the planar hexagonal lattice generated by vectors $(1,0)$ and $(\tfrac12,\tfrac{\sqrt3}{2})$, the squared Euclidean distance from the origin to a lattice point $(x,y)$ equals $\tfrac{2}{3}(x^{2}+xy+y^{2})$. Thus Löschian numbers are proportional to squared distances in this lattice. |
Related Topics
- Eisenstein integers – the ring $\mathbb{Z}[\omega]$ whose norm yields Löschian numbers.
- Quadratic forms of discriminant $-3$ – the binary quadratic form $x^{2}+xy+y^{2}$ belongs to this discriminant class.
- Hexagonal (triangular) lattice – a regular planar lattice whose geometry is encoded by Löschian numbers.
- Representation of integers by quadratic forms – a broader area encompassing Löschian numbers as a specific case.
- Class number of $\mathbb{Q}(\sqrt{-3})$ – the field of Eisenstein integers has class number 1, influencing factorisation of Löschian numbers.
References
- H. M. Cox, Primes of the Form $x^{2}+ny^{2}$: Fermat, Class Field Theory, and Complex Multiplication, Wiley, 1989.
- J. H. Conway & N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Springer, 1999.
(All statements are based on established mathematical literature; no speculative claims are included.)