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


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.


  • 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


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Cite this paper

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.

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