Carlo Sanna

Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132

Volume 18, 2012, Number 3, Pages 35—37

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## 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

APASanna, Carlo (2012). A new elementary proof of the inequality * φ*(

*n*) >

*π*(

*n*), Notes on Number Theory and Discrete Mathematics, 18(3), 35-37.

Sanna, C. “A New Elementary Proof of the Inequality * φ*(

*n*) >

*π*(

*n*).” Notes on Number Theory and Discrete Mathematics 18, no. 3 (2012): 35-37.

Sanna, Carlo . “A New Elementary Proof of the Inequality * φ*(

*n*) >

*π*(

*n*).” Notes on Number Theory and Discrete Mathematics 18.3 (2012): 35-37. Print.