We describe a new connection between the dilogarithm function and the solutions of Pell’s equation \(x^2-ny^2 = \pm 1\). For each solution x, y to Pell’s equation, we obtain a dilogarithm identity whose terms are given by the continued fraction expansion of the associated unit \(x+y\sqrt{n} \in {\mathbb {Z}}[\sqrt{n}]\). We further show that Ramanujan’s dilogarithm value-identities correspond to an identity for the regular ideal hyperbolic hexagon.

中文翻译：

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连续分数收敛下Pell方程解的对数恒等式

我们描述了对数函数和Pell方程\（x ^ 2-ny ^ 2 = \ pm 1 \）的解之间的新连接。对于Pell方程的每个解x，  y，我们获得一个对数恒等式，其项由关联单元\（x + y \ sqrt {n} \ in {\ mathbb {Z}} [\ sqrt {n}] \）。我们进一步表明，拉马努詹的对数值身份与正则理想双曲六边形的身份相对应。