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
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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

References

  1. Alpoge, L. (2015) van der Waerden and the primes, Amer. Math. Monthly, 122 (8), 784–785.
  2. Darmon, H. & Merel, L. (1997) Winding quotients and some variants of Fermat’s last theorem, J. Reine Angew. Math., 490, 81–100.
  3. Dickson, L. E. (1971) History of the Theory of Numbers, Chelsea, New York.
  4. Erdős, P. (1938) Über die Reihe Σ 1/p , Mathematica (Zutphen), B7, 1–2.
  5. Euler, L. (1737/1744) Variae observationes circa series infinitas, Com. Acad. Scient. Petropl., 9, 160–188.
  6. Furstenberg, H. (1955) On the infinitude of primes, Amer. Math. Monthly, 62, 353.
  7. Granville, A. (2017) Squares in atithmetic progressions and infinitely many primes, Amer. Math. Monthly, 124 (10), 951–954.
  8. Heath, T. L. (1908/1956) The Thirteen Books of Euclids Elements, Vol. 2, University Press, Cambridge, 2nd ed. reprinted by Dover, New York.
  9. Neukirch, J. (1992/1999) Algebraische Zahlentheorie, Springer-Verlag, Berlin, English translation Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften 322.
  10. Roth, K. F. (1953) On certain sets of integers, J. London Math. Soc., 28, 104–109.
  11. Szemerédi, E. (1969) On sets of integers containing no four elements in arithmetic progression, Acta Math. Acad. Sci. Hungar., 20, 89–104.
  12. Szemerédi, E. (1975) On sets of integers containing no k elements in arithmetic progression, Acta Arith., 27, 199–245.

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Cite this paper

APA

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.

Chicago

Seki, Shin-ichiro. “Valuations, Arithmetic Progressions, and Prime Numbers.” Notes on Number Theory and Discrete Mathematics 24, no. 4 (2018): 128-132, doi: 10.7546/nntdm.2018.24.4.128-132.

MLA

Seki, Shin-ichiro. “Valuations, Arithmetic Progressions, and Prime Numbers.” Notes on Number Theory and Discrete Mathematics 24.4 (2018): 128-132. Print, doi: 10.7546/nntdm.2018.24.4.128-132.

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