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Bounded-box reductions in the Subbarao-Warren problem for unitary perfect numbers

This paper advances the Subbarao-Warren problem on unitary perfect numbers by employing a bounded-box reduction to eliminate five impostor kernels via a three-filter certificate and providing verified finite frontiers for the remaining auxiliary set $H_{even}$ , thereby narrowing the search to a specific divisor-level problem involving cyclotomic values without yet proving finiteness.

Original authors: Tom Maciejewski

Published 2026-05-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Tom Maciejewski

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The Hunt for a "Perfect" Number

Imagine you are looking for a very special kind of number, called a Unitary Perfect Number (UPN).

The Rule: A number is "perfect" if the sum of its special "unitary" parts equals exactly twice the number itself.
The Mystery: Mathematicians have only found five of these numbers in all of history (the largest one is a massive number with 24 digits).
The Question: Are there any more? Or is the list of five complete?

This paper is a massive, high-tech search mission to prove that no new numbers exist, or at least to narrow down the search so tightly that finding one would be nearly impossible.

The Strategy: The "Bounded Box" and the "Impostors"

The authors, led by Tom Maciejewski, decided to stop guessing and start systematically checking every possible candidate within a specific "box" of rules.

Think of building a UPN like building a house. You start with a seed (a specific odd number) and add bricks (prime factors) to it.

The Known Houses: We know of two specific "blueprints" (called kernels) that successfully built the known perfect houses (specifically the numbers 90 and the huge 5th number).
The Impostors: The authors ran a computer simulation to see if there were any other blueprints that looked like they could build a perfect house but hadn't been found yet. They found five "impostor" blueprints. These look promising but, according to the paper, are actually fake.

The Goal: Prove that these five impostor blueprints can never actually build a perfect house.

The Three Filters: How They Catch the Fakes

To prove the impostors are fake, the authors built a three-stage security checkpoint (a "certificate") that every candidate number must pass. If a number fails any stage, it's thrown out.

Filter Z (The Zsigmondy Gate):
- Analogy: Imagine a bouncer checking IDs. This filter checks if the numbers involved have a "primitive" prime factor that is too old or too new to be allowed. If the math doesn't line up with a famous theorem (Zsigmondy's), the candidate is kicked out immediately.
- Result: This caught about 495 impostors.
Filter N (The Non-3-Higgs Witness):
- Analogy: This is a background check. The paper defines a special club called "3-Higgs primes." If a number's family tree includes a "bad apple" (a prime that isn't in the 3-Higgs club), the whole number is disqualified.
- Result: This was the most effective filter, catching 1,614 impostors. It even worked on numbers that weren't fully factored yet, by finding just one "bad apple" in their partial family tree.
Filter O (The 2-Adic Budget Overshoot):
- Analogy: Imagine you have a strict budget for a construction project. As you add more bricks (factors), you calculate the "cost" in a specific currency (powers of 2). If the cost of the bricks exceeds the budget allowed by the seed number, the project is impossible.
- Result: This caught the remaining 10 stubborn impostors that the other filters missed.

The Verdict: By the time they finished checking numbers up to a massive limit (10,000), every single one of the five impostor blueprints was proven fake. The only blueprints left that might work are the two we already know about.

The Remaining Mystery: The "Even" Set ( $H_{even}$ )

Even though they cleared the impostors, there is one tricky group of numbers left, called $H_{even}$ .

These are numbers where every single prime factor passes the "3-Higgs" background check.
The authors suspect this group is finite (meaning it stops growing after a certain point), but they can't prove it yet.
They did a rigorous count up to 50,000 and found that there are at most 272 candidates left in this group.
They proved that if this group is infinite, it would have to be "thin" (very sparse), but they couldn't prove it stops completely.

The "Missing Link": Why It's Hard to Finish

The paper admits it hasn't solved the whole mystery yet. Here is the final hurdle:

To prove there are no more perfect numbers, they need to prove that for very large numbers, the math simply cannot work out.
They identified a specific mathematical "gap." It's like knowing a bridge is too weak to hold a truck, but not having the engineering formula to prove exactly where it breaks for every possible truck size.
They propose a new conjecture (a guess based on strong evidence) called the "Divisor Log-Mass Conjecture." If this conjecture is true, the mystery is solved. If it's false, the hunt continues.

Summary of Results

Impostors Eliminated: They rigorously proved that the five "fake" blueprints for new perfect numbers are impossible within the tested range.
The Search Space Shrunk: They reduced the problem to checking a very small, specific list of about 272 "suspicious" numbers.
Reproducibility: They released all their code, data, and verification logs so anyone can run the tests again and see the results for themselves.
The Final Boss: They didn't prove the conjecture is true, but they reduced the problem to a single, precise mathematical question about how prime numbers are distributed in specific algebraic formulas.

In short: The paper is a massive cleanup operation. It swept the floor, threw out all the fake leads, and left the mathematicians with a very small, very specific pile of dust to investigate. They know exactly what that dust is; they just need a new tool to prove it's not a diamond.

Technical Summary: Bounded-Box Reductions in the Subbarao–Warren Problem for Unitary Perfect Numbers

Problem Statement
The paper addresses the Subbarao–Warren problem concerning Unitary Perfect Numbers (UPNs). A positive integer $n$ is a UPN if $\sigma^*(n) = 2n$ , where $\sigma^*$ sums the unitary divisors of $n$ . Only five examples are currently known: 6, 60, 90, 87360, and 146361946186458562560000. The strict conjecture posits that these five are the only UPNs.

The defining equation for a UPN $n = 2^a \prod p_i^{e_i}$ (with $p_i$ odd) is the "full balance":
$(2^a + 1) \prod (p_i^{e_i} + 1) = 2^{a+1} \prod p_i^{e_i}$
Here, $2^a + 1$ acts as the "seed factor" initiating a dependency cascade. Every odd prime divisor of a UPN must be a "3-Higgs" prime (a prime $p$ where every prime factor of $p-1$ is also 3-Higgs, with exponents in the prime factorization of $p-1$ bounded by 3). The paper investigates whether new UPNs can exist by analyzing "impostor kernels"—source components in the odd dependency graph that are not part of the known examples but satisfy initial structural constraints.

Methodology
The author employs a combination of rigorous computational enumeration and analytic number theory, structured around a "bounded-box" approach and a three-filter certification system.

Bounded Enumeration Box ( $B$ ): The search is restricted to a specific domain: primes $p \le 2000$ , exponents $e \le 6$ , $p^e \le 10^9$ , and strongly connected components (SCCs) of size $\le 6$ . Within this box, the author enumerates source-compatible SCCs.
Three-Filter Certificate: For each candidate kernel and seed exponent $a$ $a$ , the paper applies three rigorous filters to eliminate the possibility of a UPN:
- Filter Z (Zsigmondy/Higgs): Uses Zsigmondy's theorem to detect exponent obstructions where primitive prime divisors of $p^e+1$ violate 3-Higgs constraints.
- Filter N (Non-3-Higgs Witness): Checks divisors $m$ of $a$ (where $a/m$ is odd). If $2^m+1$ contains a non-3-Higgs prime factor, the seed $2^a+1$ inherits this factor, invalidating the candidate. This filter utilizes partial factorizations from FactorDB and recursive Pratt-tree verifications.
- Filter O (2-adic Budget Overshoot): Calculates the cumulative 2-adic valuation $v_2$ required by the odd components. If the sum of $v_2(p^e+1)$ exceeds the available budget $a+1$ , the candidate is eliminated.
Analysis of the Set $H_{even}$ : The paper defines $H_{even}$ as the set of even integers $m$ such that every prime divisor of $2^m+1$ is 3-Higgs. The existence of a new UPN with a nonsquarefree odd part is linked to the finiteness of this set. The author reduces the problem to the "prime case" ( $m=2p$ ) and analyzes the structural properties of these candidates using Aurifeuillean factorizations and deep Pratt-tree descents.

Key Contributions and Results

Elimination of Impostor Kernels: Within the bounded box $B$ $B$ , the author identifies five "impostor kernels" (in addition to the two known kernels 32 and 54) that pass initial cheap filters. Using the three-filter certificate, the paper proves that for all seed exponents $1 \le a \le 10,000$ $1 \leq a \leq 10, 000$ , no UPN exists with an odd-component structure refining these impostor kernels.
- Statistics: Out of 2,119 candidates in this range, 495 were eliminated by Filter Z, 1,614 by Filter N, and 10 by Filter O.
Rigorous Bounds on $H_{even}$ : The paper provides verified bounds for the set $H_{even}$ $H_{e v e n}$ :
- $|H_{even} \cap [2, 40000]| \le 201$ (with 191 undecided candidates).
- $|H_{even} \cap [2, 50000]| \le 272$ (with 262 undecided candidates).
- These bounds are rigorous, relying on APR-CL primality verification transcripts for large witness primes (up to 3053 digits) and conservative handling of partial factorizations.
Structural Reductions:
- Theorem 7: Proves that $H_{even}$ is finite if and only if the subset of prime cases $H_{even}^{prime} = \{2p \in H_{even} : p \text{ is prime}\}$ is finite. Furthermore, $|H_{even}| \le 4^{|H_{even}^{prime}|}$ .
- Theorem 21: Establishes an unconditional "power-saving thinness" bound for $H_{even}$ (i.e., the count grows as $O(X^{1-\eta})$ ), though this does not prove finiteness.
Identification of the Analytic Obstacle: The paper isolates the precise barrier to proving finiteness. While Ford's theorem provides a global density bound for 3-Higgs primes, it is insufficient at the scale of primitive divisors of cyclotomic values $\Phi_{4p}(2)$ . The remaining task is a divisor-level problem: proving that for large $p$ , the primitive prime divisors of $2^{2p}+1$ cannot all be 3-Higgs.

Significance and Claims
The paper explicitly states that it does not prove the strict Subbarao–Warren conjecture (that only five UPNs exist). Instead, it achieves a "bounded-box elimination" and a "verified finite frontier."

Reduction to a Single Analytic Question: The work reduces the problem of finding new UPNs (within the bounded box) to the finiteness of $H_{even}$ , which in turn hinges on the behavior of prime divisors of $\Phi_{4p}(2)$ .
Precise Target for Future Work: The author identifies the "missing theorem" as a divisor-level equidistribution or log-mass bound for the prime factors of $\Phi_{4p}(2)$ . Specifically, the paper formulates Conjecture 23 (Hybrid semigroup-friable shifted-prime conjecture) and Conjecture 29 (Divisor mod-16 equidistribution) as the necessary conditions to close the problem.
Reproducibility: The paper emphasizes that all computational results are reproducible. It provides a factor cache, APR-CL verification transcripts, and Python scripts (impostor_certificate.py, verify_Heven_rigorous.py) to verify the elimination of impostor kernels and the bounds on $H_{even}$ .

In summary, the paper rigorously eliminates all potential "impostor" sources for new UPNs up to a seed exponent of 10,000 within a defined search space and reduces the remaining open problem to a specific, well-defined analytic conjecture regarding the prime divisors of cyclotomic polynomials.