Abstract
We solve the open problem of Y. Z. Gürtaş of factoring polynomials , where
are Chebyshev polynomials of the second kind. While the factors of
were known, the pattern of their mapping to
and
was not, and it was intriguing. Each factor is a polynomial of the form
with integer coefficients whose roots are
. We then consider the analogous problem for Chebyshev polynomials of the first kind by deriving a factorization of
. This factorization enables the method used to solve the open problem to be applied to this one, resulting in a more direct proof.
ACKNOWLEDGMENTS
I thank the referees and Susan Jane Colley for their helpful suggestions. I am grateful to the College of Engineering & Computer Science at The Australian National University for research support.