On Dris Conjecture about Odd Perfect Numbers

TL;DR

This paper investigates Dris's conjecture on odd perfect numbers, deriving new inequalities involving their prime factorization and providing bounds on their components.

Contribution

It establishes a lower bound for N^2 in terms of π and α, advancing understanding of the structure of hypothetical odd perfect numbers.

Findings

N^2 > (1/2)π^γ, with γ=max{ω(n)-1, α}

ω(n) ≥ 9, the number of distinct prime factors of n

Provides new inequalities related to the Euler form of odd perfect numbers

Abstract

The Euler's form of odd perfect numbers, if any, is $n=\pi^{\alpha}N^2$, where $\pi$ is prime, $(\pi,N)=1$ and $\pi\equiv \alpha \equiv 1 \pmod{4}$. Dris conjecture states that $N>\pi^{\alpha}$. We find that $N^2>\frac{1}{2}\pi^{\gamma}$, with $\gamma=max\{\omega(n)-1,\alpha\}$; $\omega(n)\geq 9$ is the number of distinct prime factors of $n$.

Equations

$$n={\pi }^{\alpha }{N}^2$$

$$n=\frac{\sigma ({\pi }^{\alpha })}{2}\sigma (N^2)$$

$$N^2=A\frac{\sigma ({\pi }^{\alpha })}{2}and\sigma (N^2)=A{\pi }^{\alpha }$$

$$N^2=A\frac{\sigma ({\pi }^{\alpha })}{2}$$

$$A=\frac{\sigma (N^2)}{{\pi }^{\alpha }}$$

$${\pi }^{\alpha }=\sigma (N^2)=\sigma (\prod _{k=1}^{\omega (N)}{q}_k^{2{\beta }_k})=\prod _{k=1}^{\omega (N)}\sigma (q_k^{2{\beta }_k})=\prod _{k=1}^{\omega (N)}{\pi }^{{\delta }_k}$$

$$\alpha \ge \omega (n)-1$$

$$N^2=\frac{1}{2}\sigma ({\pi }^{\alpha })>\frac{1}{2}{\pi }^{\alpha }$$

$$N^2\ge \frac{3}{2}\sigma ({\pi }^{\alpha })>\frac{3}{2}{\pi }^{\alpha }$$

$$(\neg a\wedge d)\vee (a\wedge b\wedge c)\vee (b\wedge c\wedge d)$$

$$\left\{\begin{array}[c]{c}lemma\hskip 2.84544pt2.2:(a\implies b\land c)\\ lemma\hskip 2.84544pt2.3:(\neg a\implies d)\end{array}\right.$$

$$[\neg a\vee (b\wedge c)]\wedge (a\vee d)$$

$$(\neg a\wedge d)\vee (a\wedge b\wedge c)\vee (b\wedge c\wedge d)$$

$$N^2>\frac{3}{2}{\pi }^{\omega (n)-1}>\frac{1}{2}{\pi }^{\omega (n)-1}$$

$$N^2>\frac{1}{2}{\pi }^{\gamma },where\gamma =max\{\omega (n)-1,\alpha \}$$

Full text

1 INTRODUCTION

Without explicit definitions all the numbers considered in what follows must be taken as strictly positive integers. Let $\sigma(n)$ be the sum of the divisors of a number $n$; $n$ is said to be perfect if and only if $\sigma(n)=2n$. The multiplicative structure of odd perfect numbers, if any, is

[TABLE]

where $\pi$ is prime, $\pi\equiv\alpha\equiv 1\pmod{4}$ and $(\pi,N)=1$ (Euler, cited in [3, p. 19]); $\pi^{\alpha}$ is called the Euler’s factor. From equation (1) and from the fact that the $\sigma$ is multiplicative, it results also

[TABLE]

where $\sigma(N^{2})$ is odd and $2\|\sigma(\pi^{\alpha})$. Many details concerning the Euler’s factor and $N^{2}$ are given, for example, in [8][9][10][5][2]. Regarding the relation between the magnitudo of $N^{2}$ and $\pi^{\alpha}$ it has been conjectured by Dris that $N>\pi^{\alpha}$ [4]. The result obtained in this paper is a necessary condition for odd perfection (Theorem 2.1) which provides an indication about Dris conjecture. Indicating with $\omega(n)$ the number of distinct prime factors of $n$, we prove that (Corollary 2.3):

$(i)$          $N^{2}>\frac{1}{2}\pi^{\gamma}$, where $\gamma=max\{\omega(n)-1,\alpha\}$

Since $\omega(n)\geq 9$ (Nielsen, [6]), it follows:

$(i)_{1}$        $N^{2}>\frac{1}{2}\pi^{8}$; this improves the result $N>\pi$ claimed in [1] by Brown in his approach to Dris conjecture.

Besides

$(i)_{2}$       If $\omega(n)-1>2\alpha$, then $N>\pi^{\alpha}$

so that

$(i)_{3}$       If $\omega(n)-1>2\alpha$ for each odd perfect number $n$, then Dris conjecture is true.

Now, some questions arise: $\omega(n)$ depends on $\alpha$? Is there a maximum value of $\alpha$? The minimum value of $\alpha$ is $1$? The only possible value of $\alpha$ is 1 (Sorli, [7, conjecture 2]) so that Dris conjecture is true? Without ever forgetting the main question: do odd perfect numbers exist?

2 THE PROOF

Referring to an odd perfect number $n$ with the symbols used in equation (1), we obtain:

Lemma 2.1**.**

If $n$ is an odd perfect number, then

[TABLE]

Proof.

From equation (2) and from the fact that $(\sigma(\pi^{\alpha}),\pi^{\alpha})=1$, it follows

[TABLE]

where $A$ is an odd positive integer given by

[TABLE]

∎

In relation to the odd parameter $A$ in Lemma 2.1, we give two further lemmas:

Lemma 2.2**.**

If $A=1$, then $\alpha\geq\omega(n)-1\hskip 2.84544ptand\hskip 2.84544ptN^{2}>\frac{1}{2}\pi^{\alpha}$

Proof.

Let $q_{k},k=1,2,...,\omega(N)=\omega(N^{2})$, are the prime factors of $N^{2}$; from hypothesis and from (4) we have

[TABLE]

in which $\alpha=\sum_{k=1}^{\omega(N)}\delta_{k}\geq\sum_{k=1}^{\omega(N)}1_{k}=\omega(N)$.

Since $\omega(n)=\omega(N)+1$, it results

[TABLE]

Besides, from Equation (3) it follows

[TABLE]

∎

Lemma 2.3**.**

If $A>1$, then $N^{2}>\frac{3}{2}\pi^{\alpha}$

Proof.

From Equation (3) it results $A\geq 3$. Thus

[TABLE]

∎

The following theorem summarizes a necessary condition for odd perfection.

Theorem 2.1**.**

If $n$ is an odd perfect number, then

[TABLE]

where: $a\cong(A=1)$,$\neg a\cong(A>1)$, $b\cong(\alpha\geq\omega(n)-1)$, $c\cong(N^{2}>\frac{1}{2}\pi^{\alpha})$, $d\cong(N^{2}>\frac{3}{2}\pi^{\alpha})$

Proof.

We combine Lemmas 2.2 and 2.3 setting

[TABLE]

where, since it cannot be $A<1$, it is $(a)\cong(A=1)$ and $(\neg a)\cong(A>1)$. One obtains from (5)

[TABLE]

which is equivalent to

[TABLE]

∎

Considering cases in which the necessary condition for odd perfection (6) is false, we obtain the following corollaries:

Corollary 2.1**.**

If $n$ is an odd perfect number, then $N^{2}>\frac{1}{2}\pi^{\alpha}$

Proof.

We have

(7)    $(\neg c\land\neg d)(\cong\ N^{2}<\frac{1}{2}\pi^{\alpha})\implies n$ is not an odd perfect number

From the contrapositive formulation of (7) it follows the proof. ∎

Corollary 2.2**.**

If $n$ is an odd perfect number, then

[TABLE]

Proof.

We have

(8)    $(\neg b\land\neg d)(\cong\ N^{2}<\frac{3}{2}\pi^{\omega(n)-1})\implies n$ is not an odd perfect number

From the contrapositive formulation of (8) it follows the proof. ∎

Combining these two corollaries, we have

Corollary 2.3**.**

If $n$ is an odd perfect number, then

[TABLE]

Proof.

Immediate. ∎