Mathematics student and teacher, sequence enthusiast, and Yellow-Pig-seeker.
Some of my mathematical interests:
- Finite group theory
- Linear algebra
- Recreational math
- Programming in Mathematica
Memos to self
- in Cheb basis
- in Cheb basis
- Constant coefficients of each
- No. nonzero terms of each
Portolan numbers and Chebyshev Polynomials
Currently seeking solutions to the following equation (see A277402):
where are integers.
When , we get this pretty thing, which I think is new to the literature:
I conjecture that are the only solutions, but I don't have proof yet. Solutions occur when ,
Where is a Chebyshev polynomial of the first kind and is the minimal polynomial (over the integers) of .
Update 26 Nov.
I've now proven the the conjecture, but using the Conway-Jones theory of trigonometric diophantine equations with no reference to Chebyshev polynomials. I must say, the eventual proof was kind of a let-down; I think the Chebyshev angle would show something deeper.
The portolan number is equal to the following:
- is the number of intersection points within the polygon,
- is the number of lines that end at opposite corners instead of on sides (I call these "vanishing endpoints"),
- for fixed (n,k) is the number of points on the interior where exactly i lines concur (),
- is the number of points on *border* of the polygon where exactly i lines concur. (i=1 means one line intersecting with an edge or corner)
- is the number of vanishing endpoints where, in addition, i lines concur. ().
For the case , for . It would be nice to know for which other n this is the case, as it simplifies the calculation immensely.