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The abc conjecture implies infinitely many non-Wieferich places for fixed bases in number fields

Hester Graves, Benjamin Weiss

TL;DR

The paper extends the conjectural growth of non-Wieferich primes from the rational setting to number fields by proving analogues for non-Wieferich places base $a$ in Galois and imaginary quadratic fields under the abc conjecture. The authors develop norm-based machinery for algebraic integers, leverage cyclotomic factorization via $C_{n,a}$, $D_{n,a}$ and their primed variants, and establish lower bounds on these norms to identify primes in arithmetic progressions with $Nm( rak p)mod k$ constraints. They prove two growth results: a first bound yielding $ gtr rac{ ext{log } x}{ ext{log log } x}$ non-Wieferich places with $Nm( rak p)mod k=1$, and a stronger version implying $ gtr ext{log } x$ non-Wieferich places under appropriate hypotheses, with an explicit exception list in the imaginary quadratic case. Overall, the work generalizes and sharpens prior prime-based results to a broad class of number fields and algebraic bases, underpinning a deeper parallel between Wieferich phenomena and abc-type growth in arithmetic geometry.

Abstract

Silverman showed that, assuming the $abc$ conjecture, there are $\gg \log x$ non-Wieferich primes base $a$ less than $x$ \cite{silverman}, for all non-zero $a$. This inspired Graves and Murty \cite{Graves}, Chen and Ding \cite{Chen1} \cite{Chen2}, and then Ding \cite{Ding} to find growth results, assuming the $abc$ conjecture, for non-Wieferich primes $p$ base $a$, where $p \equiv 1 \pmod{k}$ for integers $k \geq 2$. In light of Murty, Srinivas, and Subramani's recent work on `the Wieferich primes conjecture' and Euclidean algorithms in number fields \cite{murty}, number theorists need results on non-Wieferich places in number fields. We prove analogues of the results of Graves \& Murty and Ding, and show Ding's result holds for all bases $a$ in all imaginary quadratic fields' rings of integers, with $31$ explicitly listed exceptions. Along the way, we generalize useful results on rational integers to algebraic integers.

The abc conjecture implies infinitely many non-Wieferich places for fixed bases in number fields

TL;DR

The paper extends the conjectural growth of non-Wieferich primes from the rational setting to number fields by proving analogues for non-Wieferich places base in Galois and imaginary quadratic fields under the abc conjecture. The authors develop norm-based machinery for algebraic integers, leverage cyclotomic factorization via , and their primed variants, and establish lower bounds on these norms to identify primes in arithmetic progressions with constraints. They prove two growth results: a first bound yielding non-Wieferich places with , and a stronger version implying non-Wieferich places under appropriate hypotheses, with an explicit exception list in the imaginary quadratic case. Overall, the work generalizes and sharpens prior prime-based results to a broad class of number fields and algebraic bases, underpinning a deeper parallel between Wieferich phenomena and abc-type growth in arithmetic geometry.

Abstract

Silverman showed that, assuming the conjecture, there are non-Wieferich primes base less than \cite{silverman}, for all non-zero . This inspired Graves and Murty \cite{Graves}, Chen and Ding \cite{Chen1} \cite{Chen2}, and then Ding \cite{Ding} to find growth results, assuming the conjecture, for non-Wieferich primes base , where for integers . In light of Murty, Srinivas, and Subramani's recent work on `the Wieferich primes conjecture' and Euclidean algorithms in number fields \cite{murty}, number theorists need results on non-Wieferich places in number fields. We prove analogues of the results of Graves \& Murty and Ding, and show Ding's result holds for all bases in all imaginary quadratic fields' rings of integers, with explicitly listed exceptions. Along the way, we generalize useful results on rational integers to algebraic integers.

Paper Structure

This paper contains 9 sections, 17 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Background
  3. The $abc$ conjecture and the $abc$ conjecture for number fields
  4. Wieferich places
  5. Useful Results on Norms of Algebraic Integers
  6. The norm of $a^n -1$ and its factors
  7. First Growth Result on non-Wieferich places
  8. Norms of Cyclotomic Polynomials in Number Fields
  9. A Stronger Growth Result on Weiferich Places

Key Result

Theorem 1

Suppose that $K$ is a Galois number field, that $\varepsilon >0$, that $k \in {\mathbb{Z}}^+$ , and that $a$ is a non-zero algebraic integer (i.e. $a \in {\mathcal{O}}_K \setminus \{0\}$) that is not a root of unity. If one assumes the abc conjecture for $K$, then there are $\gg_{\varepsilon, K, a,k

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Conjecture 1
  • Conjecture 2
  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 26 more