Roots with product equal to 1

Let $n \ge 2$ be a positive integer. Then let $y_n = n + \sqrt{n^2-1}.$ There are integers $a_n,c_n$ such that $y_n^{2020} = c_n + a_n\sqrt{n^2-1}.$ Then if $b_n = c_n-na_n,$ it is easy to check that $y_n^{2020} = a_ny_n + b_n.$ Now notice that $1/y_n = n-\sqrt{n^2-1},$ the ( Galois ) conjugate of $y_n,$ and conjugating both sides immediately shows that $(1/y_n)^{2020} = a_n(1/y_n) + b_n.$ (This can be checked directly as well, without any field theory.)

Since there are infinitely many $y_n,$ there are infinitely many equations of the form $x^{2020} = a_n x + b_n$ which have two distinct real roots whose product is 1. It's possible that some of these equations are the same (i.e. $a_m = a_n, b_m=b_n$ for some $m,n$ ), but any particular equation of this type has finitely many roots, hence finitely many $y_n$ as roots, so there must be infinitely many distinct equations of this type as well.