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November 2016 Balancing non-Wieferich primes in arithmetic progression and $abc$ conjecture
Sudhansu Sekhar Rout
Proc. Japan Acad. Ser. A Math. Sci. 92(9): 112-116 (November 2016). DOI: 10.3792/pjaa.92.112

Abstract

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 $\mathbf{Q}(\sqrt{2})$, there are infinitely many balancing non-Wieferich primes. In particular, under the assumption of the $abc$ conjecture for the number field $\mathbf{Q}(\sqrt{2})$ there are at least $O(\log x/{\log \log x})$ such primes $p \equiv 1(\mathrm{mod}\ k)$ for any fixed integer $k> 2$.

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Sudhansu Sekhar Rout. "Balancing non-Wieferich primes in arithmetic progression and $abc$ conjecture." Proc. Japan Acad. Ser. A Math. Sci. 92 (9) 112 - 116, November 2016. https://doi.org/10.3792/pjaa.92.112

Information

Published: November 2016
First available in Project Euclid: 2 November 2016

zbMATH: 06705716
MathSciNet: MR3567596
Digital Object Identifier: 10.3792/pjaa.92.112

Subjects:
Primary: 11A41 , 11B25 , 11B39

Keywords: $abc$ conjecture , arithmetic progression , Balancing number , Wieferich prime

Rights: Copyright © 2016 The Japan Academy

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Vol.92 • No. 9 • November 2016
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