Let us recapitulate what we have been able to prove so far:

(A) Sorli's conjecture implies the OPN conjecture, provided we could rule out the specific condition

p = q = 5

as detailed out in this post. (Note that p = 5 is the smallest possible Euler prime since p == 1 (mod 4).)

(B) The abundancy index value of 2 (i.e. the classical concept of number perfection) provides a "natural bifurcation" between EPNs and OPNs, in the following sense:

(1) Mersenne primes are congruent to 3 modulo 4, while "the" Euler primes are congruent to 1 modulo 4.

(2) If we can prove (A) (i.e. by ruling out p = q = 5), then we will be able to show the following (full) characterization theorem for perfect numbers:

"Enhanced Euclid-Euler Model for Perfect Numbers":

If M = (p^a)(n^2) is a perfect number with gcd(p, n) = 1, then

(a) M is even if and only if a = 1 and n < p^a.

(b) M is odd if and only a > 1 and p^a < n.

Finally, I give you some inequality plots (again from WolframAlpha) which I hope will clarify why it cannot happen that p = q = 5 (and therefore why p is different from q, in general):

Inequality Plot for (p - 1)(q - 1) < (p + 1)(q - 2), 2q >= p + 5, q = p

Inequality Plot for (p - 1)(q - 1) < (p + 1)(q - 2), 2q >= p + 5, q > p

Inequality Plot for (p - 1)(q - 1) < (p + 1)(q - 2), 2q >= p + 5, p < q

Inequality Plot for (p - 1)(q - 1) < (p + 1)(q - 2), 2q >= p + 5

Inequality Plot for (p - 1)(q - 1) < (p + 1)(q - 2), q = p

Inequality Plot for (p - 1)(q - 1) < (p + 1)(q - 2), q > p

Inequality Plot for (p - 1)(q - 1) < (p + 1)(q - 2), q < p

Inequality Plot for (p - 1)(q - 1) < (p + 1)(q - 2)

Again, I am "heuristically" claiming that:

CONJECTURE I: There is a one-to-one correspondence between the Euler primes and the OPNs.

In effect, Conjecture I is equivalent to the following related (albeit more general) conjecture:

CONJECTURE II: All squares are solitary numbers.

Please see these links for an attempt to prove Conjecture II:

[1] http://groups.google.com/group/sci.math/msg/c9901d8a2e03e8f4

[2] http://groups.google.com/group/sci.math/browse_thread/thread/a68aa1190a8dbbd7/c9901d8a2e03e8f4#c9901d8a2e03e8f4

These two links are archived in:

[3] http://upforthecount.com/math/abundance.html

You might also be interested in:

[4] http://upforthecount.com/math/perfect.html

In particular, note from [1] and [2] that an initial check done by Robert Israel (from Vancouver, Canada) using Maple did not produce any counterexamples to conjecture II in the range

1 <= A < B <= 300000

where

I(A^2) = Sigma(A^2)/A^2 == Sigma(B^2)/B^2 = I(B^2).

I finished typing this past 10:30 PM (Manila time)

**[last edited at 1:05 AM]**. That's all for now. Will try to resume working on clarifying the salient points by tomorrow.