Carlo Sanna
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 18, 2012, Number 3, Pages 35—37
Download full paper: PDF, 131 Kb
Details
Authors and affiliations
Carlo Sanna
Universitá degli Studi di Torino, Italy
Abstract
In this paper we provide a new elementary proof that the inequality φ(n) > π(n) holds for all integers n ≥ 91, an old result of L. Moser. Our proof is based on Bonse’s Inequality. This makes it somewhat simpler than Moser’s proof, which in turn relies on Bertrand’s Postulate.
Keywords
- Arithmetic functions
- Inequalities
AMS Classification
- 11A25
References
- Archibald, C., Bertrand’s Postulate, Scripta Mathematica, Vol. 1, 1945, 109–120.
- Bonse, H., Uer eine bekannte Eigenschaft der Zahl 30 und ihre Verallgemeinerung, Arch. Math. Phys., Vol. 12, 1907, 292–295.
- Moser, L., A theorem on the distribution of primes, Amer. Math. Monthly, Vol. 56, 1949, 624–625.
- Moser, L., On the equation φ(n) = π (n) Pi Mu Epsilon J., 1951, 101–110.
- Ramanujan, S., A proof of Bertrand’s Postulate, Journal of the Indian Mathematical Society, Vol. 11, 1919, 181–182.
Related papers
Cite this paper
Sanna, Carlo (2012). A new elementary proof of the inequality φ(n) > π (n), Notes on Number Theory and Discrete Mathematics, 18(3), 35-37.