Balancing non-Wieferich primes in arithmetic progression and \(abc\) conjecture. (English) Zbl 1419.11032

Rout, Sudhansu Sekhar

Proc. Japan Acad., Ser. A 92, No. 9, 112-116 (2016).

Summary: In this note, we shall define the balancing Wieferich prime which is an analogue of the famous Wieferich primes. We prove that, under the \(abc\) conjecture for the number field \(\mathbb{Q}(\sqrt{2})\), there are infinitely many balancing non-Wieferich primes. In particular, under the assumption of the \(abc\) conjecture for the number field \(\mathbb{Q}(\sqrt{2})\) there are at least \(O(\log x/{\log \log x})\) such primes \(p \equiv 1\pmod k\) for any fixed integer \(k> 2\).

MSC:

11B83	Special sequences and polynomials
11B25	Arithmetic progressions
11A41	Primes

Keywords:

balancing number; Wieferich prime; arithmetic progression; \(abc\) conjecture