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Some New Results on Odd Perfect Numbers - A Summary

Let $N = p^k m^2$ be a hypothetical odd perfect number with special prime $p$ satisfying $p \equiv k \equiv 1 \pmod 4$ and $\gcd(p, m) = 1$.  Note that, since $\gcd(p, m) = 1$ and $p$ is (the special) prime, then $p^k \neq m$.  (This also follows from the fact that prime powers are deficient, contradicting $N$ is perfect.)  By trichotomy, either $(p^k < m) \oplus (m < p^k)$ is true, where $\oplus$ denotes exclusive-OR.  (That is, $A \oplus B$ is true if and only if exactly one of $A$ or $B$ holds.)

We will denote the classical sum of divisors of the positive integer $z$ by $\sigma(z)=\sigma_1(z)$, and the abundancy index of $z$ by $I(z)=\sigma(z)/z$. Furthermore, we will denote the deficiency of $z$ by $D(z)=2z-\sigma(z)$, and the aliquot sum of $z$ by $s(z)=\sigma(z)-z$.

Here is a summary of some new results on odd perfect numbers, which were realized by the author on May 5, 2023:

  • If $p < m$, then the quantity $m^2 - p^k$ is not a square. (Kindly note the contrapositive.)
  • If $m < p$, then the following statements hold:
    • Descartes's conjecture holds (i.e. $k = 1$).
    • Dris conjecture (i.e. $p^k < m$) is false.
    • The quantity $m^2 - p^k$ is a square.
    • The square root of the non-Euler part $m^2$ is almost perfect.
  • Define the following GCDs:
    • $G=\gcd\left(\sigma(p^k),\sigma(m^2)\right)$
    • $H=\gcd\left(m,\sigma(m^2)\right)$
    • $I=\gcd\left(m^2,\sigma(m^2)\right)$
          where we know that $I=\sigma(m^2)/p^k = {m^2}/(\sigma(p^k)/2)$.
  • Dris proved in February 10, 2022 that the chain of divisibility conditions
    • $G \mid H \mid I$
          holds. Later on, he realized that we do in fact have

    • $G=\gcd\left(\sigma(p^k)/2,\sigma(m^2)/p^k\right)=\sigma(p^k)/2=\gcd(G,I)$
    • $H=\gcd\left(m,\sigma(m^2)/p^k\right)=m=\gcd(H,I)$.
  • In particular, the divisibility constraint $\sigma(p^k)/2 \mid m$ holds.
  • The divisibility condition $\sigma(p^k) \mid 2m$ is equivalent to $m \mid \sigma(m^2)$.
  • ( To be continued$\ldots$ )