Valuations, arithmetic progressions, and prime numbers

Shin-ichiro Seki
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 24, 2018, Number 4, Pages 128–132
DOI: 10.7546/nntdm.2018.24.4.128-132
Full paper (PDF, 156 Kb)

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Authors and affiliations

Shin-ichiro Seki
Mathematical Institute, Tohoku University
6-3, Aoba, Aramaki, Aoba-Ku, Sendai, 980-8578, Japan

Abstract

In this short note, we give two proofs of the infinitude of primes via valuation theory and give a new proof of the divergence of the sum of prime reciprocals by Roth’s theorem and Euler–Legendre’s Theorem for arithmetic progressions.

Keywords

• Infinitude of primes
• Sum of prime reciprocals
• Valuations
• Arithmetic progressions

2010 Mathematics Subject Classification

• 11A41
• 11B25

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