Euler–Euclid’s type proof of the infinitude of primes involving Möbius function

Romeo Meštrović
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 20, 2014, Number 4, Pages 33—36
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Authors and affiliations

Romeo Meštrović
Maritime Faculty, University of Montenegro
Dobrota 36, 85330 Kotor, Montenegro

Abstract

If we suppose that S = {p1, p2, …, pk} is a set of all primes, then taking x = p1p2pk + 1 into a formula due to E. Meissel in 1854 gives
(p1 − 1)(p2 − 1)…(pk − 1) = 0.
This obvious contradiction yields the infinitude of primes.

Keywords

  • Euclid’s theorem
  • Infinitude of primes
  • Euclid’s proof
  • Euler’s proof(s)
  • Möbius inversion formula
  • Meissel formula

AMS Classification

  • Primary: 11A41
  • Secondary: 11A51, 11A25

References

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  2. Dickson, L. E., History of the Theory of Numbers, Vol. I. Divisibility and Primality, Carnegie Institution ofWashington, 1919, 1920, 1923. [Reprinted Stechert, New York, 1934; Chelsea, New York, 1952, 1966, Vol. I.]
  3. Euler, L., (1734/35), De summis serierum reciprocarum, Comment. Acad. Sci. Petropol., Vol. 7, 1740, 123–134. [In Opera omnia, I.14, 73–86, Teubner, Lipsiae et Berolini, 1924.]
  4. Euler, L., (1736), Inventio summae cuiusque seriei ex dato termino generale (posthumuous paper), Comment. Acad. Sci. Petropol., Vol. 8, 1741, 9–22. [In Opera omnia, I.14, 108–123, Teubner, Lipsiae et Berolini, 1924].
  5. Euler, L., (1737), Variae observationes circa series infinitas, Comment. Acad. Sci. Petropol., Vol. 9, 1744, 160–188. [In Opera omnia, I.14, 216–244, Teubner, Lipsiae et Berolini, 1924.]
  6. Heath, T. L., The Thirteen Books of Euclid’s Elements, Vol. 2, University Press, Cambridge, 1908; 2nd ed. reprinted by Dover, New York, 1956.
  7. Kummer, E. E., Neuer elementarer Beweis des Satzes, dass die Anzahl aller Primzahlen eine unendliche ist, Monatsber. Preuss. Akad. Wiss., Berlin 1878/9, 777–778. [Collected Papers II, 669–670, Springer, Berlin-Heidelberg, 1975.]
  8. Meissel, E., Observationes quaedam in theoria numerorum, J. Reine Angew. Math., Vol. 48, 1854, 301–316.
  9. Meštrović, R., Euclid’s theorem on the infinitude of primes: A historical survey of its proofs (300 B.C.–2012), 66 pages, 2012, preprint arXiv:1202.3670v2[math.HO]
  10. Narkiewicz, W., The Development of Prime Number Theory: from Euclid to Hardy and Littlewood, Springer Monographs in Mathematics, Springer–Verlag, Berlin, 2000.
  11. Pollack, P., Not Always Burried Deep: Selections from Analytic and Combinatorial Number Theory, American Mathematical Society, 2009.
  12. Ribenboim, P., The little book of bigger primes, Second edition, Springer–Verlag, New York, 2004.
  13. Sándor, J., Crstici, B., Handbook of Number Theory II , Kluwer Academic Publishers, Dordrecht, 2004.
  14. Shapiro, H. N., Introduction to the Theory of Numbers, John Wiley & Sons, New York, 1983.

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Cite this paper

APA

Meštrović, R. (2014). Euler–Euclid’s type proof of the infinitude of primes involving Möbius function Notes on Number Theory and Discrete Mathematics, 20(4), 33-36.

Chicago

Meštrović, Romeo. “Euler–Euclid’s Type Proof of the Infinitude of Primes Involving Möbius Function.” Notes on Number Theory and Discrete Mathematics 20, no. 4 (2014): 33-36.

MLA

Meštrović, Romeo. “Euler–Euclid’s Type Proof of the Infinitude of Primes Involving Möbius Function.” Notes on Number Theory and Discrete Mathematics 20.4 (2014): 33-36. Print.

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