Natural Numbers associated with the Eisenstein Integers

 My number associated with my diurnal age today, 26908, is a member of OEIS A118886:

A118886

Numbers expressible as \(x^2 + x \cdot y + y^2, 0 \leq x \leq y \), in two or more ways.

In the case of 26908, there are three ways viz. 

• \(x=4 \) and \(y=162\) so that \(4^2+4 \cdot 162+162^2=26908\)
• \(x=62\) and \(y=124\) so that \(62^2+62 \cdot 124+124^2=26908\)
• \(x=74\) and \(y=114\) so that \(74^2+74 \cdot 114+114^2=26908\)

As stated in the OEIS comments, the numbers in this sequence represent squares of distances between two points in the triangular lattice in two or more non-trivially different ways. The triangular lattice being referred here is that formed by the so-called Eisenstein integers. Such integers are complex numbers of the form:$$z=a+ b \omega\\ \text{ where } a \text{ and } b \text{ are integers and }\\ \omega=\frac{-1+i \sqrt{3}}{2}=e^{2\pi i/3}$$Thus we have:$$a^2+ab+b^2=(a-\omega)(b-\omega^2) \text{ because }1+\omega+\omega^2=0$$Numbers of the form \(a^2-ab+b^2 \) represent the squared modulus of an Eisenstein integer:

\(|a+b\omega|^2=(a-\frac{1}{2}b)^2+\frac{3}{4}b^2=a^2-ab+b^2\)

Figure 1: from Wikipedia

Thus both numbers of the form \(a^2+ab+b^2 \) and \(a^2-ab+b^2 \) are associated with Eisenstein integers. Looking at the latter type of number we can write:$$ \begin{align} a^2+ab+b^2 &= a^2-2ab+b^2+3ab\\&=(a-b)^2+(\sqrt{3ab})^2\\&=(a-b-i\sqrt{3ab})(a-b+\sqrt{3ab}) \\&=(a-b-i\sqrt{3} \sqrt{ab})(a-b+i\sqrt{3} \sqrt{ab}) \\ &=(a-b-(2\omega+1)\sqrt{ab})(a-b+(2\omega+1)\sqrt{ab}) \\&=(a-b-\sqrt{ab}-2\sqrt{ab} \cdot \omega)(a-b+\sqrt{ab}+2\sqrt{ab} \cdot \omega) \\&=(c_1-d\omega)(c_2+d\omega)\end{align}$$which represents the product of two Eisenstein integers which could be evaluated for 26908 since we know three different sets of values for \(a\) and \(b\). For example, for the case of \(a=4 \) and \(b=162\), we have \( \sqrt{ab}=18\sqrt{2}\). There's a lot more that could be said here of course but the important takeaway is the association with the Eisenstein integers.

Getting back to the original OEIS A118886, we find that there are numbers that are of the form \(x^2+xy+y^2\) in eight or more ways. Such numbers include 53599, 63973, 74347, 84721 and 105469. Let's take the last number as an example. The \(x,y\) values associated with this number are (15, 317), (33, 307), (53, 295), (100, 263), (108, 257), (145, 227), (153, 220) and (187, 188).

Using a Jupyter notebook, I was able to find these numbers, in the range up to a little beyond one million, that can be represented in nine or more ways:

157339, 229957, 256711, 306397, 356083, 375193, 427063, 447811, 472017, 505141, 520429, 548359, 554827, 593047, 604513, 612157, 629356, 654199, 654493, 689871, 696787, 730639, 738283, 760627, 770133, 803257, 810901, 831649, 849121, 852943, 872053, 883519, 894691, 902629, 919191, 919828, 956137, 966511, 967603, 1013467, 1018381, 1026844, 1067857, 1101373, 1125579, 1173991, 1204567

Here is a permalink to the code being used but it will time out if run in SageMathCell. Further investigation yielded the numbers shown below that can be represented in ten or more ways. However, now that the list was considerably shorter, it was possible to test each of these numbers to see how many representations are possible and these numbers are shown in brackets after each number. As can be seen, the maximum is twelve in the range up to a little over one million.

375193 (12), 447811 (12), 520429 (12), 593047 (10), 696787 (12), 730639 (12), 738283 (12), 810901 (12), 831649 (12), 849121 (12), 883519 (12), 956137 (12), 966511 (12), 967603 (10), 1013467 (12), 1018381 (12), 1067857 (12), 1101373 (12), 1125579 (12)

SageMathCell makes it easy to determine the number of possible pairs for each number. It's just necessary to tset up the equation. Let's use 1125579 as an example:$$x^2 + x \cdot y + y^2 =1125579$$Here is a permalink to the code that solves this Diophantine equation and the output is:

Figure 2 shows the graph of this elliptic function and there are exactly twenty four points on this graph with coordinates that are integers. Why 24 when we have only 12 above? Well, a restriction had been applied such that \(x \leq y\) but no such restriction need apply for this ellipse. The case of A = (422, 785) and B = (785, 422) is shown below.

Figure 2: using GeoGebra