I was playing around with an idea that there might be some insight into Fermat primes by looking at products of complex solutions of polynomials of the form x^{2^n)+1=0, and Chebyshev polynomials came up as I was looking at the exact values of those roots.¹² As I recall, looking at finding half angle sines and cosines of increasing fractions, I ended up seeing Chebyshev polynomials emerging from the results.
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1. I may have this confused with my similar investigations into Mersenne primes and x^p-1=0.
2. My hypothesis that I could find factors of a Fermat number with n > 5 by multiplying the roots together and setting x = 2 failed on writing a program to actually check the result, but looking back on my memories of doing this, I may have made an error.
Chebyshev polynomials have as roots nth roots of unity, so of course these are going to show up. It's one way to define them.
https://en.wikipedia.org/wiki/Chebyshev_nodes
The nth roots of unity are incredibly well studied, and some of that stemmed in the 1700-1800s on trying to factor things. The entire field of analytic numbers theory has taken these ideas to incredible (think decades of study and research to be state of the art) depths.
More explicitly: Chebyshev polynomials are what you get when you take trigonometric polynomials for a periodic interval or Laurent polynomials on the unit complex circle and project onto a diameter of the circle.