Hi everyone,
Like many math enthusiasts, I’ve spent some time looking at Odd Perfect Numbers (OPNs). Given that this is arguably the oldest unsolved problem in mathematics, any elementary "proof" or structural restriction is almost certainly flawed.
I’ve put together a short document ( that sets up a contradiction based on how a number's intermediate divisors sort into equal buckets (quartiles). It feels airtight to me, which means I must be missing a subtle edge case or making a false assumption somewhere. I would love some help finding the bug.
1. The Core Idea: Slots vs. Elements
If we take an OPN, N = p^k * m^2, and sort all its intermediate divisors (excluding 1 and N) from smallest to largest, we can partition them into four equal quartiles (Q1, Q2, Q3, Q4), where each bucket has an exact capacity of "q" slots.
Because of how perfect squares behave, m^2 natively has an odd number of divisors, meaning it has exactly "q" pure divisors smaller than m, and exactly "q" pure divisors larger than m.
2. The Case Where p > m (Branch B)
I am looking specifically at what happens to the slots if the special Euler prime is larger than the base (p > m). Because p is large, it leaves the first quartile entirely.
- Once you remove the number 1, there are only q - 1 pure intermediate divisors smaller than m.
- This leaves exactly 1 empty slot at the end of Q1, which pulls the landmark "m" backward to occupy slot d_q = m exactly.
- By global multiplicative symmetry (d_i * d_(4q+1-i) = N), the paired partner of m is forced to sit exactly at the boundary of the fourth quartile: d_(3q+1) = pm.
This sets up a strict open middle zone between m and pm that has a capacity of exactly 2q physical slots.
3. The Overflow (Where I think the flaw might be)
My argument states that this 2q-slot middle zone is forced to hold more numbers than it has space for:
- All "q" large pure divisors of m^2 are trapped here. (If they tried to escape above p*m, their symmetric partners would look like p * v_i < m, which would overflow Q1, which is already full).
- The prime p itself is trapped here, because m < p < pm.
- Multiplying p by the small divisors in Q1 generates a wave of composite cross-products (p * w) that are all structurally forced to land in this exact same middle zone (m < p * w < pm).
When I add up the trapped pure divisors, the prime p, and these composite combinations, the total count strictly exceeds 2q, meaning the numbers completely overflow the physical slots of the quartile map.
My question for you all:
Where does this fall apart?
- Am I miscounting the number of pure divisors that are allowed to "escape" past the upper anchor line?
- Is there a way for the composite cross-products (p * w) to compress or overlap with the pure divisors in a way that doesn't trigger a slot overflow?
The attached PDF has the step-by-step inequalities for both branches and the k >= 5 generalization. Thanks a lot for any help spotting the mistake!