Loeschian Numbers

Loeschian numbers are numbers of the form \(x^2+xy+y^2\) where \(x\) and \(y\) are integers. I came across these today when looking for interesting properties associated with my diurnal age of 26575 days. The Wikipedia article states that:

They are a set of whole numbers, including zero, and having prime factorisation in which all primes congruent to 2 mod 3 have even powers (there is no restriction of primes congruent to 0 or 1 mod 3).

Now \(26575 =5^2 \times 1063\) and we see that \(5 \equiv 2 \hspace{-4pt} \mod{3}\) is raised to an even power while \(1063 \equiv 1 \hspace{-4pt} \mod{3}\). These numbers are relatively frequent. For example of the first 1000 integers, 277 (or 27.7%) of them are Loeschian. Some have only one representation such as \(26575 = 15^2+15 \times 155+155^2\) while others have more than one. For example:$$ \begin{align} 637 &=4^2+4 \times 23+ 23^2\\ &=7^2+7 \times 21+21^2 \\ &=12^2+12 \times 17+ 17^2 \\931&=1^2+1 \times 30+ 30^2 \\&=14^2+14 \times 21+21^2\\ &=9^2+ 9 \times 25+25^2 \end{align}$$It was in this context that the famous taxi cab number 1729 popped up again. I've written about this number before in a post titled The Original Taxi Cab Number in a New Light on December 21st 2019. In that post, I listed several of its properties but not the fact that it is a member of OEIS A198775:

A198775

Numbers having exactly four representations by the quadratic form \(x^2+xy+y^2\) with \(0 \leq x \leq y\).

We find that the first member of this sequence is 1729:

1729, 2821, 3367, 3913, 4123, 4459, 4921, 5187, 5551, 5719, 6097, 6517, 6643, 6916, 7189, 7657, 8029, 8113, 8463, 8827, 8911, 9139, 9331, 9373, 9709, 9919, 10101, 10507, 10621, 10633, 11137, 11284, 11557, 11739, 12369, 12649, 12691, 12901, 13237, 13377, ...

It has the following representations:$$ \begin{align} 1729 &= 23^2 +23 \times 25+25^2  \\&=3^2+3 \times 40+40^2  \\ &=15^2+15 \times 32+32^2 \\  &=8^2+ 8 \times 37+37^2 \end{align}$$

The Loeschian numbers are named after August Lösch whose Wikipedia entry remarks:

Overall, Lösch made a plenitude of significant findings in the world of economics, but his main contributions were to regional economics, specifically, pioneering the location theory, spatial equilibrium analysis and hierarchical spatial systems displaying a hexagonal pattern.

Figure 1 shows the triangular or, when combined into groups of six, the hexagonal lattice formed by the Eisenstein integers which Lösch must have used in his economic analysis. 

Figure 1

It turns out the Loeschian numbers are the norms of the Eisenstein integers. In mathematics, Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are complex numbers of the form:$$ \begin{align} z &= x + y\omega \text{ where }x \text{ and } y \text{ are integers }\\ \text{ and where }\omega &= \frac{-1 + i \hspace{2pt} \sqrt{3}}{2} = e^{i\frac{2\pi}{^3}} \end{align}$$ The 2-norm of an Eisenstein integer is just its squared modulus, and is given by:$$ \begin{align} \left|x + y\;\!\omega\right|^2 \,&= \, (x - \tfrac{1}{2} y)^2 + \tfrac{3}{4} y^2 \, \\ &= \, x^2 - xy + y^2 \end{align} $$It can be seen that we have a \(-xy\) instead of a \(+xy\) term but then again \(x\) and \(y\) are no longer restricted to being positive. For example, if we allow \(x\) and \(y\) to be negative as well as positive, then 26575 can be written as:$$ \begin{align} 26575 &=15^2+15 \times 155+155^2 \\ &=155^2-155 \times 170+170^2 \end{align}$$So that will do it for now but there is clearly much more that could be said about Loeschian numbers. More at a later date.