**Bounds for the Abundancy Indices of (p^k) and (m^2)**when N = (p^k)(m^2) is an OPN

with Euler prime p

(#) 1 < I(p^k) < 5/4 < 8/5 < I(m^2) < 2 < 57/20 < I(p^k) + I(m^2) < 3

($) 0 < (p^k)/(m^2)I(p^k) <= 2/3 < 3 <= (m^2)/(p^k)I(m^2) < (p^k)/(m^2)I(p^k) + (m^2)/(p^k)I(m^2)

(%) 0 < 1 < I(p^k) < 5/4 < 2*SQRT(5)/10 < I(p^k)I(m) < 2

(&) 11/3 <= (p^k)/(m^2)I(p^k) + (m^2)/(p^k)I(m^2)

The order relations ($), (%) and (&) were all derived from the following two observations:

(1) Divisibility/Congruence Tricks

gcd(p^k, Sigma(p^k) = 1 implies p^k divides Sigma(m^2). This means that Sigma(m^2)/p^k is an integer.

Additionally, since squares are never perfect, and also because m^2 is a divisor of N which is perfect (and hence, m^2 is deficient), therefore:

Sigma(m^2) is always odd. Now, p^k is an odd prime power (since N is odd). Thus, Sigma(m^2)/p^k is odd.

Let h := Sigma(m^2)/p^k. Suppose h = 1.

Because of the relation Sigma(p^k)Sigma(m^2) = 2(p^k)(m^2), we have:

2/h = Sigma(p^k)/m^2

Thus, we have the simultaneous Sigma relations:

(*) Sigma(p^k) = 2m^2

(**) Sigma(m^2) = p^k

However, according to the paper titled “Some New Results on Odd Perfect Numbers” by G. G. Dandapat,

J. L. Hunsucker and Carl Pomerance: No OPN satisfies both (*) and (**).

Thus, h = Sigma(m^2)/p^k = [(m^2)/(p^k)]I(m^2) >= 3.

(2) Inequality/Numerical Tricks

gcd(m, n) = 1 if and only if I(m)I(n) = I(mn)

What happens when equality does not hold?

We have: I(mn) < I(m)I(n)

In particular, when m = n, then we have:

I(m^2) < [I(m)]^2

But from the appropriate order relation in (#) we get:

8/5 < [I(m)]^2

Taking square roots:

2*SQRT(10)/5 < I(m) (Note that: SQRT(10) is approximately 3.162277660168379, which is bigger than, yet surprisingly proximate, to PI.)

Computing 2*SQRT(10)/5, this is approximately 1.2649110640673515.

But I(p^k) < 5/4 = 1.25.

Thus, we have I(p^k) < I(m). (Please do note that it was ALMOST A MISS.)

Consider the equation Sigma(p^k) = m.

Sigma(p^k) = 1 + p + p^2 + ... + p^k == 1 + 1 + 1 + ... + 1 (mod 4)

since the Euler prime p == 1 (mod 4)

== k + 1 (mod 4) == 2 mod (4) since k == 1 (mod 4)

Thus m == 2 mod 4, a contradiction to the assumption that N is an OPN.

**Nov 7, 2010 : (The equation I(p^k) = m is not equivalent to**

**the equation**

**Sigma(p^k) = m. However, a simpler**

**argument**

**will be to consider that if**

**p = m, then**

**N = (p^k)(m^2) = (p^(k + 2)),**

**a prime-power and which is**

**known to be deficient,**

**whereas**

**we assumed N is an OPN.)**

Consequently, we have two cases for further study:

Case 1: Sigma(p^k) < m

Case 2: m < Sigma(p^k)

Let's consider Case 2 first.

Since m < Sigma(p^k) implies m < (5/4)p^k, let's consider a "mathematical model for OPN(s)" that is simpler and better:

Since prime powers are always deficient, m is not equal to p. Consider Case 2 in light of this observation. ( e.g. Suppose m < p ( <= p^k < Sigma(p^k) ). )

Since (p^k)/(m^2)I(p^k) <= 2/3 and I(p^k) > 1, then this means that:

(p^k)/(m^2) < 2/3 < 1.

In particular, p <= p^k < (2/3)(m^2) < (2/3)p^2 < p^2.

That is, p^(1) <= p^(k) < p^2.

Consequently, 1 <= k < 2.

In particular, since we know a priori that k must be congruent to 1 modulo 4, it follows that k = 1.

It follows that, if m < p^k and N = (p^k)(m^2) is an OPN, then k = 1 (e.g. p || N). (This was conjectured by Ronald Sorli in 2003.) We will show that this will then lead to a contradiction.

We are assuming that m < p ( and of course, p <= p^(k) ).

But I(p) = Sigma(p)/p, which means that:

m < p < Sigma(p) < Sigma(m) < 2m

since p and m are deficient.

**But we also know that:**

**Sigma(p)/m <= 2/3. --> Retracted.**

**What we do know is that:**

**[Sigma(p)]/(m^2) <=**

**[Sigma(p^k)]/(m^2) = [(p^k)/(m^2)]I(p^k) <= 2/3.**

**Of course, since**

**[Sigma(p)]/(m^2) < [Sigma(p)]/m, the required conclusion does NOT follow.**

Thus, we have:

m < p < Sigma(p) < Sigma(m) < 2m

which implies that:

m < Sigma(p) < 2m

from which it follows that:

1 < Sigma(p)/m < 2

This contradicts:

Sigma(p)/m <= 2/3.

This leaves us to consider Case 1: Sigma(p^k) < m (in light of the

simplified “mathematical model” p < m.)

Assume Case 1 is true. Then it would imply that (p^k)/(m) < 1. Equivalently, (m)/(p^k) > 1.

Therefore, we now have the following theorem:

**:**

__Theorem [Arnie Dris, 2010]__**If N = {p^(k)}{m^(2)} is an OPN, then p^(k) < m. --> Retracted.**

__PostScript:__When I completed my masters thesis in 2008 (it was titled "Solving the Odd Perfect Number Problem - Some Old and New Approaches"), I conjectured that:

p^k < m

on the basis of the *very limited* information that p^k < m^2 "might have" followed from I(p^k) < I(m^2). Indeed, I was under the naive assumption that the "divisibility constraint" gcd(p, m) = 1 together with I(p^k) < I(m^2) induced some sort of "order property" between p^k and m^2.