发表机构
University of Bucharest; Simion Stoilow Institute of Mathematics of the Romanian Academy（布加勒斯特大学; 罗马尼亚科学院西蒙·斯托伊洛瓦数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为Pell方程最小解构造固定直线程序，利用几何级数恒等式和加权二进制编码，以94或98次操作计算，并证明最小基为2X1(X1+1)-1。
AI 中文摘要
对于非平方数$d\ge 2$，我们构造了固定的直线程序来计算方程$x^2-dy^2=1$的最小非平凡解$(X_1,Y_1)$，该程序使用加法、截断减法、乘法、整数除法、幂运算和取余运算。一个几何级数恒等式从Pell解初始段坐标和恢复$(X_1,Y_1)$，而无需知道求和了多少个解。加权二进制编码使这些和可通过算术计算访问。在复用计算值的计数操作下，一个程序使用94次操作，中间位长以$2^{2^{2^{O(d)}}}$为界。额外四个操作得到一个98次操作的程序，其界为$2^{2^{O(d)}}$。Hua的界给出相应的103次和107次操作的程序，并在这些大小界中将$O(d)$替换为$O(\sqrt d\log d)$。一旦基本解已知，生成函数公式通过提取其坐标作为基$b$的数字，在额外21次操作内计算第$n$个正解。我们证明$2X_1(X_1+1)-1$是使两个公式对所有$n\ge1$都有定义且正确的最小整数基。
英文摘要
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 $n\ge1$.
Comments27 pages
