2026/07/30 by Daniel Evans
Mathematics · #math.NT #msc:11J72 #msc:11A41 #msc:11N13 #msc:11C08
arxiv created 2026/07/30 · arxiv updated 2026/07/31
Finite logarithms of non-zero rational numbers can be defined in the "poor man's adèle ring" \mathcal A by Fermat quotients modulo sufficiently large primes. This ring contains ℚ and outside 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≡ 1\bmod m by relating Fermat quotients to values of cyclotomic polynomials and their logarithmic derivatives. As an application we show that, subject to the abc-conjecture, finite logarithms cannot be quadratic irrational in \mathcal A in an appropriate sense.