Year: 2025
Type: article
Abstract: We discuss arithmetic questions related to the ‘poor man’s adèle ring’ $$\mathcal {A}$$ A whose elements are encoded by sequences $$(t_p)_p$$ ( t p ) p indexed by prime numbers, with each $$t_p$$ t p viewed as a residue in $$\mathbb {Z}/p\mathbb {Z}$$ Z / p Z . Our main theorem is about the $$\mathcal {A}$$ A -transcendence of the element $$(F_p(q))_p$$ ( F p ( q ) ) p , where $$F_n(q)$$ F n ( q ) (Schur’s q-Fibonacci numbers) are the (1, 1)-entries of $$2\times 2$$ 2 × 2 -matrices $$\begin{aligned} \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ q & 0 \end{pmatrix} \begin{pmatrix} 1 & 1 \\ q^2 & 0 \end{pmatrix} \cdots \begin{pmatrix} 1 & 1 \\ q^{n-2} & 0 \end{pmatrix} \end{aligned}$$ 1 1 1 0 1 1 q 0 1 1 q 2 0 ⋯ 1 1 q
Source: The Ramanujan Journal
Authors: Florian Luca, Wadim Zudilin
Institutions: Stellenbosch University, Radboud University Nijmegen
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Funder: European Research Council
OpenAlex ID: W4411401832