AI 中文总结
该文在$abc$猜想成立前提下，将有限对数无理性的已有结果扩展到素数受限为$p\equiv 1\bmod m$的算术级数情形，还证明其无法为二次无理数。
AI 中文摘要
非零有理数的有限对数可通过足够大素数模的费马商在“穷小子阿代尔环”${\boldsymbol{\textit{\mathcal A}}}$中定义，该环包含${\boldsymbol{\mathbb{Q}}}$。Matsusaka和Seki已证明，除平凡情形外，${\boldsymbol{\mathcal A}}$中的有限对数不能取非零有理数值；Silverman的定理则表明，在$abc$猜想成立的前提下，这些有限对数非零。我们通过关联费马商与分圆多项式及其对数导数，将上述结果扩展到形如$p\equiv 1\bmod m$的算术级数限制的素数情形。作为应用，我们证明在$abc$猜想成立的条件下，${\boldsymbol{\mathcal A}}$中的有限对数在合适意义下不能是二次无理数。
英文摘要
Finite logarithms can be defined in the ``poor man's adèle ring" $\mathcal{A}$ using Fermat quotients modulo sufficiently large primes. This ring contains $\mathbb{Q}$ and outside the trivial cases, Matsusaka and Seki have shown that finite logarithms cannot take non-zero rational values in $\mathcal{A}$. Furthermore, a theorem of Silverman shows they are not zero, assuming the $abc$-conjecture. We extend these results to primes restricted to arithmetic progressions of the form $p\equiv 1\bmod m$ by relating Fermat quotients to values of cyclotomic polynomials and their logarithmic derivatives. A signed version of the same argument shows unconditionally that the square of a finite logarithm cannot take non-zero rational values in $\mathcal{A}$. As a further application, we show that, subject to the $abc$-conjecture, finite logarithms cannot be quadratic irrational elements of $\mathcal{A}$ in Rosen's theory of finite algebraic numbers.
CommentsVersion 2: 12 pages, included theorem 1.3 on squares of finite logarithms
