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
Full paper (PDF, 140 Kb)

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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, …, p_k} is a set of all primes, then taking x = p₁p₂…p_k + 1 into a formula due to E. Meissel in 1854 gives
(p₁ − 1)(p₂ − 1)…(p_k − 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

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