Odd Perfect Number Investigation Series Day 5
Today we established a two-directional structural pressure on the remaining escape route.
Assuming an odd perfect number of Euler form
N = p^α m²,
with all prime divisors of m congruent to 2 mod 3:
✔️ If α is kept small,
the known bound Ω(N) ≥ 101 forces the number of prime factors t of m to grow, yielding a factorial-type lower bound
N ≥ p^α · 4^t · (t!)².
✔️ If α increases,
the number r of primes satisfying p² | (q² q 1) must increase,
forcing multiple primes into extremely thin residue classes (mod 3p²), again producing factorial-type growth of N.
No resolution yet.
But the space of admissible structures is provably shrinking, without computation or distributional assumptions.
#OddPerfectNumbers
#NumberTheory
#UnsolvedProblems
#AIDEproject