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User:Ethan Beihl

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Mathematics student and teacher, sequence enthusiast, and Yellow-Pig-seeker.


Some of my mathematical interests:

  • Finite group theory
  • Linear algebra
  • ALife
  • Recreational math
  • Programming in Mathematica

Memos to self

To add:

  1. in Cheb basis
  2. in Cheb basis
  3. Constant coefficients of each
  4. 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.

n-Portolan numbers

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.