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 $(u_n{)}_{n=-\mathrm{\infty }}^{\mathrm{\infty }}$ is such that every term is in $\mathbb{Z}[1\mathrm{/}b]$, 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 $m\mathbb{Z}[1\mathrm{/}b]$. 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.