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

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## Details

### 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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## Cite this paper

Seki, S. (2018). Valuations, arithmetic progressions, and prime numbers. *Notes on Number Theory and Discrete Mathematics*, 24(4), 128-132, doi: 10.7546/nntdm.2018.24.4.128-132.