As before, let N = (p^k)(m^2) be an Odd Perfect Number (OPN) with Euler prime p and gcd(p, m) = 1.

Earlier, we have seen that assuming the inequality m < p^k was true, the equality k = 1 would follow.

Just today (December 4, 2010), it seems as though (at least to myself) I am also able to prove that, by assuming the truth of the related inequality p^k < m, the same conclusion (i.e. the equality k = 1) follows.

These two implications together (of course) imply that it is indeed true that k = 1.

Thus, Sorli's conjecture on OPNs (circa 2003), which predicts that the exponent k of the Euler prime p should be unity (i.e. k = 1), can now be rightfully called a theorem.

I will be presenting the details of the proof by December 6 (Monday) [early afternoon] to a group of mathematicians (with particular specializations in algebraic number theory, elliptic curves and coding theory) at the University of the Philippines - Diliman.