Wieferich primes in number fields and the conjectures of Ankeny–Artin–Chowla and Mordell
Year: 2026
Type: article
Abstract: For a prime p ≡ 1 ( mod 4 ) , let ε = 1 2 ( t + u p ) be the fundamental unit of the real quadratic field Q ( p ) . In 1951, N. Ankeny, E. Artin, and S. Chowla asked whether p can divide u . They suggested that this can never happen and this has since been called the Ankeny–Artin–Chowla (AAC) conjecture. We show that if p is the prime above p in Q ( p ) , then the AAC conjecture is false if and only if ε p − 1 ≡ 1 ( mod p 2 ) . Thus, the AAC conjecture is related to the existence of number field analogues of Wieferich primes. Therefore, in the second part of this paper, we investigate the infinitude of Wieferich primes in number fields. Subject to Masser's abc -conjecture for number fields, we prove that for any fixed α ∈ O K ∖ { 0 } that is not a root of unity, there are infinitely many primes ideals p ⊆ O K for which α N ( p ) − 1 ≢ 1 ( mod p 2 ) . Additionally, we show under the weaker assumption that there are finitely many base-α super-Wieferich primes , and that there are infinitely many base- α non-Wieferich primes. In both cases, we obtain the lower bound # { prime ideals p : N ( p ) ≤ x and α N ( p ) − 1 ≢ 1 ( mod p 2 ) } ≫ K , α , ϵ log x log log x as x → ∞ .
Source: Journal of Number Theory
Authors: Nic Fellini, M. Ram Murty
Institution: Queen's University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,550
OpenAlex ID: W7127891853