论文标题
用二芬太汀方程的解决方案作为$ p $ - adic代数函数的定期点，iii
Solutions of diophantine equations as periodic points of $p$-adic algebraic functions, III
论文作者
论文摘要
确定了与Rogers-Ramanujan持续分数$ r（τ）$相关的某个代数函数的所有周期点。它们被证明为$ 0，\ frac {-1 \ pm \ sqrt {5}} {2} $，以及值$ r（w_d/5）$的$ \ mathbb {q} $上的偶联，其中$ w_d $是$ w_d $是一个特定的ealgebraic Integers的特定集合，划分为5的特定派别，该派别是一个PRIPER DIVIS，a divis primper a Prime primper a p Inder a a in prigence a and a a和a divis of a n a n aft a a n aft and a a和a $ k_d = \ mathbb {q}（\ sqrt {-d}）$，as $ -d $范围在所有负二次判别物上，$ \ left（\ frac {-d} {5} {5} {5} {5} \ right）= +1 $。这产生了对字段中订单$ k_d $的阶级数量的新见解。第I部分的猜想1被证明是Prime $ P = 5 $的，表明$ k_d $字段的环类字段相对高至$ 5 $，与$ \ Mathbb {q} $上生成的字段相吻合，该字段由定期点（不包括-1 $ -5 $ -ADADIC -ADIC -ADIC -ADIC ALGEBRAIC FOUNTION的定期点（不包括-1）。
All the periodic points of a certain algebraic function related to the Rogers-Ramanujan continued fraction $r(τ)$ are determined. They turn out to be $0, \frac{-1 \pm \sqrt{5}}{2}$, and the conjugates over $\mathbb{Q}$ of the values $r(w_d/5)$, where $w_d$ is one of a specific set of algebraic integers, divisible by the square of a prime divisor of 5, in the field $K_d=\mathbb{Q}(\sqrt{-d})$, as $-d$ ranges over all negative quadratic discriminants for which $\left(\frac{-d}{5}\right) = +1$. This yields new insights on class numbers of orders in the fields $K_d$. Conjecture 1 of Part I is proved for the prime $p=5$, showing that the ring class fields over fields of type $K_d$ whose conductors are relatively prime to $5$ coincide with the fields generated over $\mathbb{Q}$ by the periodic points (excluding -1) of a fixed $5$-adic algebraic function.
