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Balancing non-Wieferich primes in arithmetic progressions

  • Utkal Keshari Dutta
  • Bijan Kumar Patel
  • Prasanta Kumar RayEmail author
Article

Abstract

A prime is called a balancing non-Wieferich prime if it satisfies \(B_{p - \genfrac(){}{}{8}{p}} \not \equiv 0\pmod {p^{2}},\) where \(\genfrac(){}{}{8}{p}\) and \(B_n\) denote the Jacobi symbol and the n-th balancing number respectively. For any positive integers \(k > 2\) and \(n > 1\), there are \(\gg \log x / \log \log x\) balancing non-Wieferich primes \(p \le x\) such that \(p \equiv 1 \pmod {k}\) under the assumption of the abc conjecture for the number field \(\mathbb {Q}(\sqrt{2})\) (Proc. Japan Acad. Ser. A 92 (2016) 112–116). In this paper, for any fixed M, the lower bound \(\log x / \log \log x\) is improved to \((\log x/ \log \log x)(\log \log \log x)^{M}\).

Keywords

Balancing numbers Wieferich primes arithmetic progressions abc conjecture 

2010 Mathematics Subject Classification

11B25 11B39 11B41 

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Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  • Utkal Keshari Dutta
    • 1
  • Bijan Kumar Patel
    • 2
  • Prasanta Kumar Ray
    • 1
    Email author
  1. 1.Sambalpur UniversityBurlaIndia
  2. 2.International Institute of Information TechnologyBhubaneswarIndia

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