Some modular considerations regarding odd perfect numbers - Part II

In this article, we consider the various possibilities for p and k modulo 16, and show conditions under which the respective congruence classes for sigma(m(2)) (modulo 8) are attained, if p(k)m(2) is an odd perfect number with special prime p. We prove that 1. sigma(m(2)) equivalent to 1 (mod 8) holds only if p + k equivalent to 2 (mod 16). 2. sigma(m(2)) equivalent to 3 (mod 8) holds only if p k equivalent to 4 (mod 16). 3. sigma(m(2)) equivalent to 5 (mod 8) holds only if p + k equivalent to 10 (mod 16). 4. sigma(m(2)) equivalent to 7 (mod 8) holds only if p k equivalent to 4 (mod 16). We express gcd(m(2); sigma(m(2))) as a linear combination of m(2) and sigma(m(2)). We also consider some applications under the assumption that sigma(m(2))/p(k) is a square. Lastly, we prove a last-minute conjecture under this hypothesis.