Year: 2026
Type: preprint
Abstract: For nonsquare $d\ge 2$, we construct fixed straight-line programs for the least non-trivial solution $(X_1,Y_1)$ of $x^2-dy^2=1$, using addition, truncated subtraction, multiplication, integer division, exponentiation, and remainder. A geometric-sum identity recovers $(X_1,Y_1)$ from the coordinate sums of an initial segment of Pell solutions, without knowing how many solutions were summed. Weighted binary encodings make these sums accessible to arithmetic computation. Counting operations with reuse of computed values, one program uses $94$ operations, with intermediate bit lengths bounded by $2^{2^{2^{O(d)}}}$. Four additional operations give a $98$-operation program with the bound $2^{2^{O(d)}}$. Hua's bound gives corresponding programs with $103$ and $107$ operations and replaces $O(d)$ by $O(\sqrt d\log d)$ in these size bounds. Once the fundamental solution is known, generating-function formulas compute the $n$-th positive solution in $21$ further operations by extracting its coordinates as base-$b$ digits. We prove that $2X_1(X_1+1)-1$ is the least integer base for which both formulas are defined and correct for every $n\ge1$.
Source: arXiv (Cornell University)
Authors: Dumitru, Bogdan, Prunescu, Mihai
Institution: University of Bucharest
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214298882