Paolo Starni

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 24, 2018, Number 1, Pages 5–9

DOI: 10.7546/nntdm.2018.24.1.5-9

**Full paper (PDF, 141 Kb)**

## Details

### Authors and affiliations

Paolo Starni

*School of Economics, Management, and Statistics
Rimini Campus, University of Bologna
Via Anghera 22, 47921 Rimini, Italy
*

### Abstract

The Euler’s form of odd perfect numbers, if any, is *n* = *π ^{α}N*

^{2}, where

*π*is prime, (

*π*,

*N*) = 1 and

*π*≡

*α*≡ 1 (mod 4). Dris conjecture states that

*N*>

*π*. We find that

^{α}*N*

^{2}> 1/2

*π*, with

^{γ}*γ*= max{

*ω*(

*n*) − 1,

*α*};

*ω*(

*n*) ≥ 10 is the number of distinct prime factors of

*n*.

### Keywords

- Odd perfect numbers
- Dris conjecture

### 2010 Mathematics Subject Classification

- 11A05
- 11A25

### References

- Brown, P. (2016) A partial proof of a conjecture of Dris, http://arxiv.org/abs/1602.01591v1.
- Chen, S. C., & Luo, H. (2011) Odd multiperfect numbers, http://arxiv.org/abs/1102.4396.
- Dickson, L. E. (2005) History of the Theory of Numbers, Vol. 1, Dover, New York.
- Dris, J. A. B. (2008), Solving the odd perfect number problem: some old and new approaches, M.Sc. thesis, De La Salle University, Manila, http://arxiv.org/abs/1204.1450.
- MacDaniel, W. L., & Hagis, P. (1975) Some results concerning the non-existence of odd perfect numbers of the form
*π*^{α}M^{2β}, Fibonacci Quart., 131, 25–28. - Nielsen, P. P. (2015) Odd perfect numbers, Diophantine equations, and upper bounds, Math. Comp., 84, 2549–2567.
- Sorli, R. M. (2003) Algorithms in the study of multiperfect and odd perfect numbers, Ph.D. thesis, University of Technology, Sydney, http://epress.lib.uts.edu.au/research/handle/10453/20034.
- Starni, P. (1991) On the Euler’s factor of an odd perfect number, J. Number Theory, 37, 366–369.
- Starni, P. (1993) Odd perfect numbers: a divisor related to the Euler’s factor, J. Number Theory, 44, 58–59.
- Starni, P. (2006) On some properties of the Euler’s factor of certain odd perfect numbers, J. Number Theory, 116, 483–486.

## Related papers

- Dris, J. A. B., & San Diego, I. T. (2020). Some modular considerations regarding odd perfect numbers – Part II.
*Notes on Number Theory and Discrete Mathematics*, 26 (3), 8-24. - Dagal, K. A. P., & Dris, J. A. B. (2021). The abundancy index of divisors of odd perfect numbers – Part II.
*Notes on Number Theory and Discrete Mathematics*, 27(2), 12-19.

## Cite this paper

Starni, P. (2018). On Dris conjecture about odd perfect numbers. *Notes on Number Theory and Discrete Mathematics*, 24(1), 5-9, DOI: 10.7546/nntdm.2018.24.1.5-9.