Power residue symbols and the exponential local-global principle
Henry Robert Thackeray
TL;DR
The paper addresses the exponential local-global principle (Skolem conjecture) for second order linear recurrences with simple rational roots, aiming to establish new quadratic and degenerate cubic cases. It expresses the sequence as $u_n = b_1 c_1^n + b_2 c_2^n$, reduces the zero term condition to the congruence $C^n = B$, and employs $n$th power residue symbols together with Schinzel S77 to produce infinitely many primes with prescribed residue behaviors. A key contribution is the proof of the remaining Case 3 of the main theorem, showing the existence of infinitely many primes where $C^k$ never equals $B$ modulo those primes under a nondegeneracy condition. The results extend known quadratic cases of the Skolem conjecture and yield degenerate cubic corollaries, illustrating a method that leverages Chebotarev density type tools to study local-global phenomena in linear recurrences and potentially informing broader families of recurrences.
Abstract
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.