You add 1 to the golden ratio to get its square. How do you find higher powers?
Yatir from Israel wrote this article on numbers that can be written as $ 2^n-n $ where n is a positive integer.
A sequence of polynomials starts 0, 1 and each poly is given by
combining the two polys in the sequence just before it. Investigate
and prove results about the roots of the polys.