Power residue symbols and the exponential local-global principle

The exponential local-global principle, or Skolem conjecture, says: Suppose that $b$ is a positive integer, and that the sequence $(u_{n})_{n = -\infty}^{\infty}$ is such that every term is in $\mathbb{Z}[1/b]$, the linear recurrence $u_{n + d} = a_{1}u_{n + d - 1} + \cdots + a_{d}u_{n}$ holds for all integers $n$, and every root of $x^{d} - a_{1}x^{d - 1} - a_{2}x^{d - 2} - \cdots - a_{d}$ is nonzero and simple; then there is no zero term $u_{n}$ if and only if, for some integer $m$ that is larger than $1$ and relatively prime to $b$, every term $u_{n}$ is not in $m\mathbb{Z}[1/b]$. Particular cases of the conjecture are known, but the general conjecture is open. This paper proves some apparently new quadratic and degenerate cubic cases of the exponential local-global principle via power residue symbols. This work was presented at the Stellenbosch Number Theory Conference 2025 in January 2025 at Stellenbosch University; much of the work was also presented at the 67th Annual Congress of the South African Mathematical Society in December 2024 at the University of Pretoria.