2025/08/11 by Nic Fellini, M. Ram Murty, Fellini, Nic +1 · 1 citation
Mathematics · #11G05 (Secondary) #11N69 #11R04 (Primary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2508.08472
openalex publication_date 2025/08/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a prime p≡ 1 (\bmod4), 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 \mathfrakp is the prime above p in ℚ(√(p)), then the AAC conjecture is false if and only if εp-1 ≡ 1 (\bmod\mathfrakp2). 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∈ OK∖\0\ that is not a root of unity, there are infinitely many primes ideals \mathfrakp⊆ OK for which α^N(\mathfrakp)-1 \not≡ 1 (\bmod\mathfrakp2). 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 \mathfrakp : N(\mathfrakp)≤ x and α^N(\mathfrakp)-1\not≡ 1 (\bmod\mathfrakp2)\ ≫K, α, ε (log x)/(loglog x) as x→ ∞.