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.

Cyclotomic Polynomials

Recently I turned 26142 days and this number has the property that it is a member of OEIS A138938:

A138938

Indices k such that A019326(k)=\( \Phi_8\) is prime, where \( \Phi \) is a cyclotomic polynomial.

I've heard many times about cyclotomic polynomials over the past five years without really understanding their significance. This occasion provided an opportunity to investigate the topic further. I discovered that SageMath (which I use for most of my calculations) has a function to generate the cyclotomic polynomials. The following simple command will generate the first twenty cyclotomic polynomials:

for n in [1..20]:

    print(n,"-->",cyclotomic_polynomial(n,x))

\( \Phi_1 = x - 1\)

\( \Phi_2 = x + 1\)

\( \Phi_3 = x^2 + x + 1\)

\( \Phi_4 = x^2 + 1\)

\( \Phi_5 = x^4 + x^3 + x^2 + x + 1\)

\( \Phi_6 = x^2 - x + 1\)

\( \Phi_7 = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1\)

\( \Phi_8 = x^4 + 1\)

\( \Phi_9 = x^6 + x^3 + 1\)

\( \Phi_{10} = x^4 - x^3 + x^2 - x + 1\)

\( \Phi_{11} = x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1\)

\( \Phi_{12} = x^4 - x^2 + 1\)

\( \Phi_{13} = x^{12} + x^{11} + x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1\)

\( \Phi_{14} = x^6 - x^5 + x^4 - x^3 + x^2 - x + 1\)

\( \Phi_{15} = x^8 - x^7 + x^5 - x^4 + x^3 - x + 1\)

\( \Phi_{16} = x^8 + 1\)

\( \Phi_{17} = x^{16} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1\)

\( \Phi_{18} = x^6 - x^3 + 1\)

\( \Phi_{19} = x^{18} + x^{17} + x^{16} + x^{15} + x^{14} + x^{13} + x^{12} + x^{11} + x^{10} + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1\)

\( \Phi_{20} =  x^8 - x^6 + x^4 - x^2 + 1\)

But what are the cyclotomic polynomials, or cyclotomic polynumbers as they are also called, and why are they important? A definition of a cyclotomic polynumber is that it is an irreducible polynumber with integer coefficients, \(n=1,2,3,...\) such that:$$ 1-x^n=\prod_{d|n} \Phi_d \text{ where } d\text{ are the divisors of } n$$By irreducible is meant they cannot be factored into a product of polynomials of lesser degree. In this respect, these polynumbers are like prime numbers. The name cyclotomic arrives from the view of these polynumbers being solutions to \(1-z^n\) on the complex plane. The solutions lie on the unit circle and can be thought of as cutting up this circle into \(n\) sectors (see Figure 1 that shows the solutions to \(1-z^{10}=0\)).

Figure 1

From the earlier definition it can be seen that:$$1-z^{10}=\Phi_1 \times \Phi_2 \times \Phi_5 \times \Phi_{10} \text{ and so}$$ $$1-z^{10}=(z-1)(z+1)(z^4 + z^3 + z^2 + z + 1)(z^4 - z^3 + z^2 - z + 1)$$There are some other interesting results, including:$$ \Phi_p=1+x+x^2+...+ \,x^{p-1} \text{ where } p \text{ is prime }$$Looking at the cyclotomic polynumbers shown at the beginning of this post, it can be seen that \( \Phi_2, \Phi_3, \Phi_5, \Phi_7, \Phi_{11}, \Phi_{13}, \Phi_{17} \text{ and } \Phi_{19} \) follow this pattern. Another interesting result is that:$$ \Phi_{p^k}=\Phi_p(x^{p^{k-1}}) \text{ with } p \text{ again prime }$$An example illustrating the previous result is: $$\Phi_9=\Phi_{3^2}=\Phi_3(x^3)=1+x^3+(x^3)^2 + 1=1+x^3+x^6$$Another property of cyclotomic polynumbers is that:$$ \Phi_{pm} \Phi_m=\Phi_m(x^p) \text{ where } p \text{ is prime and gcd}(p,m)=1$$This result could be used to find \( \Phi_{60}\) in terms of lesser polynumbers because 60 = 5 x 12 and so:$$\Phi_{60}\; \Phi_{12}=\Phi_{12}(x^5) = (x^5)^4 - (x^5)^2 + 1 = x^{20}-x^{10}+1$$ $$ \Phi_{60}= \frac{x^{20}-x^{10}+1}{x^4 - x^2 + 1}=1+x^2-x^6-x^8-x^{10}+x^{14}+x^{16}$$Another useful result arising from the previous result as the case where \(p=2\) is:$$ \Phi_{2m} \; \Phi_m=\Phi_m(x^2) \text{ where } m \text{ is odd }$$ $$ \text{ Hence } \Phi_{2m}= \frac{\Phi_m(x^2)}{\Phi_m}$$As a particular example, let's work out \( \Phi_{14} \) where of course 14 = 2 x 7:$$ \Phi_{14}=\frac{\Phi_7(x^2)}{\Phi_7}=\frac{1+x^2+x^4...+ \, x^{12}}{1+x+x^2+...+ \, x^6}=1-x+x^2-x^3+x^4-x^5+x^6$$Lastly, for this post at least, it can be noted that there is a connection with Euler's totient function via the relationship:$$ \text{ degree of } \Phi_n=\phi(n) \text{ where } \phi \text{ is Euler's totient function}$$We can see this result at work if we look at say \( \Phi_{12} = x^4 - x^2 + 1\) which has degree 4 and \(\phi(12)=4 \) since the numbers that are coprime to 12 are 1, 5, 7 and 11. I'd like to thank N. J. Wildberger's Insights into Mathematics YouTube channel for providing some of the examples and information in this blog post. Here is a link to this channel's videos on Cyclotomic Polynomials:

• Computing cyclotomic polynumbers 
• Irreducibility and the Schoenemann-Eisenstein criterion
• Logical difficulties with cyclotomic fields
• Tips and tricks for computing cyclotomic polynumbers

Cyclotomic polynumbers fall into the category of algebraic number theory and what's interesting about N. J. Wildberger's approach is that he doesn't feel complex numbers (falling into the category of Complex Analysis) should be used to explain their properties. Of course, I've only scratched the surface of this topic but at least I've made a start that I can build on in later posts.