A function field variant of Pillai's problem

Abstract

In this paper, we consider a variant of Pillai's problem over function fields $F$ in one variable over $\mathbb{C}$. For given simple linear recurrence sequences $G_n$ and $H_m$, defined over $F$ and satisfying some weak conditions, we will prove that the equation $G_n-H_m=f$ has only finitely many solutions $(n,m)\in {\mathbb{N}}^2$ for any non-zero $f\in F$, which can be effectively bounded. Furthermore, we prove that under suitable assumptions there are only finitely many effectively computable $f$ with more than one representation of the form $G_n-H_m$.

13 pages

References in corpus (2)

• On a problem of Pillai with $k$-generalised Fibonacci numbers and powers of $2$
• Perfect Powers: Pillai's works and their developments

Cited by in corpus (3)

• On the size of a linear combination of two linear recurrence sequences over function fields
• Pillai's conjecture for polynomials
• $S$-unit values of $G_n+G_m$ in function fields