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Port Fillings for Primary Pseudoperfect Numbers

This paper investigates the infinitude of primary pseudoperfect numbers by introducing a "port" framework based on local residual equations and arithmetic derivatives to distinguish between fillings inherited from smaller solutions and primitive ones, thereby addressing Erdős's Problem #313 beyond the previously known computational limits.

Original authors: Han Wang

Published 2026-05-22

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Han Wang

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

Imagine you are a master architect trying to build a very specific, magical tower. The rules for building this tower are strange: you must stack blocks (which are prime numbers) in such a way that if you add up the "reciprocals" of all the blocks, plus a tiny fraction for the whole tower, the total equals exactly 1.

In the world of mathematics, these special towers are called Primary Pseudoperfect Numbers. For a long time, mathematicians knew how to build towers with up to 8 blocks, but they were stuck. They didn't know if they could build a tower with 9 blocks, or 10, or if there was an infinite supply of these magical towers.

This paper by Han Wang is like a new set of blueprints and a new construction tool that finally helps us build the 9-block and 10-block towers, and suggests a way to keep building forever.

Here is how the paper works, broken down into simple concepts:

1. The New Tool: "Ports" and "Fillings"

Think of the math problem as a game of plumbing.

The Port: Imagine a pipe with a specific shape and pressure requirement. In math terms, this is a "Port," defined by two numbers. It represents a specific equation that needs to be satisfied.
The Filling: To make the water flow perfectly (to make the equation equal 1), you need to insert a specific plug made of prime numbers. This plug is called a "Filling."

The author introduces a clever rule called the Composition Law. Think of it like a "Lego inheritance" rule. If you have a small, valid Lego structure (a small tower) and you want to build a bigger one, you don't have to start from scratch. You can attach a new "filling" to your existing "port." This paper separates fillings that are just copies of old towers (inherited) from "port-primitive" fillings, which are brand-new, unique structures that couldn't have been built by just adding a single block to a smaller tower.

2. The Big Discovery: The "Key Port"

The author focused on a specific, tricky Port (let's call it the Key Port). This port was waiting for a filling.

Discovery 1: The author found a "port-primitive" filling made of just two prime numbers ( $149 \times 3109$ ). When you attach this to the base, you get a known tower with 7 blocks.
Discovery 2 (The Big One): The author found a different, much larger port-primitive filling made of four prime numbers ( $157 \times 1979 \times 10093 \times 16879$ ).

When you attach this four-block filling to the Key Port, you get a brand new, massive tower with 9 prime factors. This is the number $N_9$ . Before this paper, no one knew if a 9-block tower existed. Now we have one.

3. The "Magic Step": From 9 to 10

Once the 9-block tower ( $N_9$ ) was built, the author checked the number right after it ( $N_9 + 1$ ).

In a lucky twist, this number turned out to be a prime number (a number that can't be divided by anything other than 1 and itself).
Because of a mathematical rule (Corollary 5.2), if you have a valid tower and the number right after it is prime, you can simply multiply them together to get a new, valid tower.
This created the 10-block tower ( $N_{10}$ ).

The paper provides a "Pocklington certificate," which is like a rigorous, unbreakable ID card proving that the 10-block tower and the prime number after the 9-block tower are real and valid.

4. The "What If" Scenario: Building Forever

The paper also asks: Can we keep building these towers forever?
The author doesn't prove this is definitely possible (that would be a huge, unsolved math problem). Instead, they say: "If we accept a specific, reasonable guess about how prime numbers are distributed, then yes, we can build infinitely many."

They propose a "Five-Splitting" hypothesis. Imagine you have a single prime block at the top of your tower. The hypothesis suggests that under certain conditions, you could theoretically replace that one block with five new, larger blocks that still satisfy the magical rules. If you could do this over and over, you would have an infinite supply of towers.

The author proves that the math allows for this splitting to happen locally (in small steps), but the final step relies on a famous, unproven guess in mathematics (the Bateman–Horn conjecture) adapted for five numbers at once.

5. The "Last Two Blocks" Trick

Finally, the paper gives a "discriminant criterion." Think of this as a quality control checklist for the very last two blocks you need to add to a tower.
Instead of guessing and checking millions of numbers, the author created a formula. If you plug your numbers into this formula, it tells you instantly if the last two blocks will work. It turns a massive search problem into a simple check: "Is this specific number a perfect square?" If yes, you might have a tower; if no, you don't.

Summary

The Goal: Build towers of prime numbers that sum to 1.
The Method: A new "Port and Filling" system that organizes how these towers are built.
The Result: Found a new 9-block tower and a new 10-block tower.
The Future: Proved that if a certain mathematical guess is true, we can build an infinite number of these towers by splitting one block into five.
The Tool: A new formula to quickly check if the last two blocks of a tower will fit.

This paper doesn't just find new numbers; it builds a new framework (the "Port" system) that makes finding these rare mathematical structures much more systematic and less like random guessing.

Technical Summary: Port Fillings for Primary Pseudoperfect Numbers

Problem Statement
The paper addresses the problem of primary pseudoperfect numbers (PPNs), defined as squarefree integers $n$ satisfying the equation:
$\frac{1}{n} + \sum_{p|n} \frac{1}{p} = 1$
Equivalently, using the arithmetic derivative $\partial(n)$ , these are squarefree solutions to $\partial(n) = n - 1$ . The existence of infinitely many such numbers is an open problem, linked to Erdős's Problem #313. While Butske, Jaje, and Mayernik established the uniqueness of PPNs for up to 8 prime factors via computation, the paper seeks to extend this search to higher layers and investigate the infinitude of the set.

Methodology: The Port Formalism
The author introduces a "local formalism" to analyze the residual equations governing PPNs.

Ports and Fillings: A "port" is defined as a pair $(R, c)$ of positive integers. A squarefree integer $B$ "fills" the port if it satisfies the defect equation:
$\Delta_{R,c}(B) := cB - R\partial(B) = 1$
This is equivalent to the reciprocal form $\sum_{q|B} \frac{1}{q} + \frac{1}{RB} = \frac{c}{R}$ .
Composition Law: The paper establishes a composition law $\Delta_{R,c}(AB) = \Delta_{RA, \Delta_{R,c}(A)}(B)$ for coprime $A, B$ . This law distinguishes between "inherited" fillings (derived from smaller PPNs) and "port-primitive" fillings (new solutions not reducible to smaller fillings of the same port).
Inheritance: The framework generalizes the known inheritance mechanism where if $K$ is a PPN and $K+1$ is prime, then $K(K+1)$ is a PPN. This is framed as a specific case of port filling where the port is $(K, 1)$ .

Key Contributions and Results

Discovery of New PPNs ( $N_9$ and $N_{10}$ ):
- The author isolates a specific "key port" $H = (113322, 797)$ , derived from the prefix $2 \cdot 3 \cdot 11 \cdot 17 \cdot 101$ .
- Two distinct port-primitive fillings of $H$ $H$ are identified:
  - $B_2 = 149 \cdot 3109$ (2 primes), yielding the known PPN $52495396602$.
  - $B_4 = 157 \cdot 1979 \cdot 10093 \cdot 16879$ (4 primes).
- The second filling generates a new PPN with nine prime factors:
  $N_9 = 113322 \cdot B_4 = 5998279018951962402$
- The primality of $N_9 + 1$ is verified using a Pocklington certificate. Since $N_9$ is a PPN and $N_9+1$ is prime, the product $N_{10} = N_9(N_9+1)$ is a PPN with ten prime factors:
  $N_{10} = 35979351189199316534587473905773572006$
Disproof of a Conjecture:
- The paper addresses the Sondow–MacMillan conjecture, which predicted the existence of exactly one PPN with nine prime factors and asserted that no further PPNs exist.
- The construction of $N_{10}$ (a 10-prime-factor PPN) disproves the clause stating "no further primary pseudoperfect numbers exist."
- The paper notes that $N_9$ satisfies the predicted residue modulo 288 ( $N_9 \equiv 258 \pmod{288}$ ), supporting the uniqueness prediction for the 9-factor case, but the existence of $N_{10}$ refutes the finiteness claim.
Algorithmic Criteria for Search:
- Last-Two-Prime Discriminant: A criterion is derived to determine if a port can be completed by exactly two primes. This reduces the problem to checking if a specific quadratic polynomial $D(t)$ takes square values within a finite interval.
- Modular Square Sieve: The paper employs modular sieving to exclude prefixes that cannot lead to valid fillings, providing a computational method to prune the search space for higher-layer PPNs.
Conditional Infinitude Criterion:
- The paper proposes a conditional proof for the infinitude of PPNs based on a "five-splitting Hardy–Littlewood–Bateman–Horn type prime-points hypothesis."
- The hypothesis posits that for a "terminal port" (where a single prime $p$ fills $(R, c)$ ), the hypersurface defined by $cx_1x_2x_3x_4x_5 - R\sum \prod_{j \neq i} x_j = 1$ contains points where all coordinates are distinct primes.
- Under this hypothesis, a terminal prime can be recursively replaced by five larger primes, generating an infinite sequence of PPNs. The author explicitly states this is not a consequence of the classical one-variable Bateman–Horn conjecture.

Significance and Scope
The paper's primary significance lies in the development of the "port formalism," which unifies the study of PPNs, Znám-type problems, and arithmetic derivatives into a single local framework.

Unconditional Results: The discovery of $N_9$ and $N_{10}$ , the verification of their properties, the Pocklington certificate for $N_9+1$ , and the discriminant criterion for the final two primes are presented as unconditional mathematical facts.
Conditional Results: The infinitude theorem is strictly conditional on Hypothesis 19.2. The author is modest regarding this, noting that no unconditional mechanism for infinite PPNs is currently known.
Computational Limits: The paper includes exploratory data for the "H6 layer" (six-prime fillings of port $H$ ) but clarifies that these are computational notes and not part of the proofs for $N_9$ or $N_{10}$ . The search for a 12-prime PPN (via $N_{10}+1$ ) is shown to fail, as $N_{10}+1$ is composite.

In summary, the paper expands the known landscape of primary pseudoperfect numbers, provides a rigorous structural framework for their generation, and offers a conditional pathway to proving their infinitude, while carefully delineating between proven theorems and conjectural hypotheses.