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Frequency Ordered Ratio Families Arising from the Factorization of $p_{m-1}+1$

This paper investigates a frequency-ordered sequence of ratio values derived from the factorization of $p_{m-1}+1$ , explaining the emergence of distinct "families" in these plots and proposing a heuristic asymptotic model based on primes in arithmetic progressions to account for the observed frequency distribution.

Original authors: Alexander R Povolotsky

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexander R Povolotsky

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 have a giant machine that takes a number, finds the prime number just before it, subtracts one, and then adds one back. Mathematically, this looks like $p_{m-1} + 1$ . If you run this machine for thousands of different numbers, you get a massive list of results.

Most of the time, these results are made up of small building blocks (prime factors). But sometimes, the machine produces a result with a giant building block that is larger than the number you started with.

This paper is about finding those special "giant block" moments, breaking them down, and sorting the leftovers. Here is the story of what the author, Alexander Povolotsky, discovered, explained in everyday terms.

1. The "Giant Block" Filter

Think of the number $p_{m-1} + 1$ as a chocolate bar. Usually, this bar is made of many small, equal-sized squares.
However, for certain specific numbers (which the author calls the "A223881 family"), the chocolate bar has one giant square that is bigger than the whole bar itself (mathematically, the largest prime factor is bigger than the index $m$ ).

When this happens, the author says: "Let's snap off that giant square."

The Giant Square: This is the largest prime factor ( $L_m$ ).
The Leftover: This is the rest of the bar ( $R_m$ ).

The paper focuses entirely on these Leftovers ( $R_m$ ).

2. The "Families" on the Graph

If you were to plot every single one of these "Leftovers" on a graph, you wouldn't see a random mess. Instead, you would see distinct lines or "families" of dots.

Why? Because the math works like a seesaw.

If the Giant Square is huge, the Leftover must be tiny.
If the Giant Square is just a little bit bigger than the starting number, the Leftover is a bit larger.

So, every time the "Leftover" happens to be the number 2, all those data points line up on one specific curve. Every time the "Leftover" is 3, they line up on a different curve. The most common curves are the ones with the smallest leftovers (2, 3, 4, etc.).

3. The "Frequency-Ordered" List

The author decided to take all these leftovers, count how often each number appears, and line them up from "Most Common" to "Least Common."

The result is a specific sequence of numbers:
2, 3, 4, 8, 6, 12, 10, 14, 15...

Why does 2 come first? It's the most common leftover.
Why does 3 come second? It's the next most common.
Why is 8 before 6? Even though 6 is smaller than 8, the number 8 appears as a leftover more often in this specific mathematical process.

It's like sorting a bag of marbles by color, but instead of red or blue, the "colors" are numbers, and you sort them by how many of each you found.

4. The "Why" (The Heuristic Model)

The paper tries to explain why some numbers are more common than others using a simple rule of thumb:

Imagine you are trying to fit a puzzle piece (the number $R$ ) into a slot.

The Fit: The number $R$ has to "fit" perfectly into the math equation. The paper suggests that numbers with fewer "fitting rules" (mathematically, numbers with a low Euler totient value, $\phi(r)$ ) are easier to find.
The Rarity: Finding a prime number is like finding a needle in a haystack. The rarer the needle, the harder it is to find.

The author proposes that the frequency of a leftover number $R$ is roughly determined by how "easy" it is to fit into the equation divided by how hard it is to find a prime number.

Small numbers (like 2, 3, 4) are easy to fit and easy to find, so they appear constantly.
Larger numbers are harder to fit and harder to find, so they appear rarely.

5. What the Computer Did

To prove this, the author wrote a computer program (using Mathematica) that ran this "chocolate bar" experiment 50,000 times.

It checked every number up to 50,000.
It snapped off the giant prime factors.
It counted the leftovers.
It confirmed that the list 2, 3, 4, 8, 6... is indeed the correct order of popularity.

6. What We Still Don't Know (Open Problems)

The paper ends by admitting that while the computer shows us the pattern, we don't have a perfect mathematical proof for why it happens exactly this way.

The Mystery: We can guess the pattern using probability (like guessing the weather), but we can't prove it with absolute certainty yet.
The Questions: Does this list ever stop changing? Are there hidden rules for when prime numbers show up as leftovers?

Summary

In short, this paper is about taking a weird mathematical machine, filtering out the results that have a "giant" part, and looking at the "small" parts that are left over. By sorting these small parts by how often they appear, the author found a beautiful, predictable pattern that links the randomness of prime numbers to a structured list of numbers. It's a bit like finding that if you sort all the different sizes of pebbles on a beach by how common they are, they form a perfect, predictable line.

Technical Summary: Frequency-Ordered Ratio Families Arising from the Factorization of $p_{m-1} + 1$

Problem Statement
This paper investigates the structural properties of the integer sequence defined by $p_{m-1} + 1$ , where $p_n$ denotes the $n$ -th prime number. The study focuses specifically on the subset of indices $m \ge 3$ belonging to the OEIS sequence A223881. This sequence is defined by the condition that the largest prime factor of $p_{m-1} + 1$ , denoted $L_m$ , exceeds $m$ (i.e., $L_m > m$ ). For most $m$ , all prime factors of $p_{m-1} + 1$ are $\le m$ ; thus, the set A223881 represents a sparse but structured collection of "exceptional" indices.

The central problem is to understand the distribution of the complementary factor $R_m = (p_{m-1} + 1) / L_m$ within this subset. Computational plots of $p_{m-1} + 1$ for $m \in \text{A223881}$ reveal a "family" phenomenon where data points cluster along distinct curves. The paper seeks to explain this clustering and derive a sequence based on the frequency of the values $R_m$ .

Methodology
The authors employ a combination of computational enumeration, heuristic probabilistic modeling, and classical number theory:

Computational Enumeration: The study generates values of $p_{m-1} + 1$ for $m$ up to $N = 50,000$ . For each $m$ , the integer is factored into $L_m R_m$ . The authors filter for cases where $L_m > m$ , collecting the resulting $R_m$ values into a multiset.
Frequency Ordering: The distinct values of $R_m$ are sorted by their frequency of occurrence (decreasing order) to form a new sequence, $S(N)$ .
Heuristic Modeling: The authors propose a probabilistic model to estimate the frequency of a specific value $r$ $r$ appearing as $R_m$ $R_{m}$ . This model relies on two conditions:
- $r$ must divide $p_{m-1} + 1$ .
- The quotient $(p_{m-1} + 1)/r$ must be prime (specifically, $L_m$ ).
Theoretical Analysis: The paper utilizes Dirichlet's theorem on primes in arithmetic progressions to estimate the density of $m$ satisfying the divisibility condition $r \mid (p_{m-1} + 1)$ .

Key Contributions and Results

Definition of the Frequency-Ordered Ratio Sequence: The paper defines a sequence $S(N)$ consisting of distinct $R_m$ values sorted by frequency. For $N=50,000$ , the initial segment of this sequence is $2, 3, 4, 8, 6, 12, 10, 14, 15, 18, \dots$ . The authors note that the initial segment appears to stabilize as $N$ increases.
Explanation of the "Family" Phenomenon: The paper clarifies that the distinct curves observed in plots of $p_{m-1} + 1$ correspond to fixed values of $R_m$ . Since $L_m > m$ is a restrictive condition, $L_m$ must be large, forcing $R_m$ to be relatively small. Consequently, points with the same small $R$ value cluster along the curve $p_{m-1} + 1 = R \cdot L_m$ . The most prominent families correspond to the most frequent values of $R_m$ .
Divisibility Theorem (Theorem 1): The paper establishes that for a fixed integer $r \ge 1$ , the count of $m \le N$ such that $r \mid (p_{m-1} + 1)$ is asymptotically $N / \phi(r)$ , where $\phi$ is Euler's totient function. This result is derived heuristically from Dirichlet's theorem.
Heuristic Asymptotic Model (Conjecture 1): Combining the divisibility density ( $1/\phi(r)$ ) with the heuristic probability that the quotient is prime ( $1/\log N$ ), the authors propose that the frequency of a specific $R_m = r$ follows the law:
$\#\{m \le N : R_m = r\} \sim \frac{CN}{\phi(r) \log N}$
where $C$ is a positive constant near 1.
Numerical Verification: Log-log analysis of empirical data up to $N=50,000$ confirms a linear relationship between $\log(\text{count}(r))$ and $\log(\phi(r))$ with a slope near $-1$, supporting the inverse-Euler-totient weighting.
Prime Values of R: The paper observes that while $R_m$ can be prime, such occurrences are rare among the most frequent values. Only $R=2$ and $R=3$ appear frequently; larger prime $R$ values occur extremely infrequently.

Significance and Claims
The paper claims to provide a new perspective on the structure of OEIS A223881 by linking the "family" phenomenon in $p_{m-1} + 1$ to the frequency distribution of the complementary factor $R_m$ . The primary significance lies in the observation that the ordering of these families is governed asymptotically by the inverse of Euler's totient function, $\phi(r)^{-1}$ .

The authors modestly frame their work as an explanation of observed computational patterns and a proposal of a heuristic model. They explicitly state that the rigorous proof of the inverse- $\phi(r)$ law remains an open problem. The paper does not claim to have proven the asymptotic distribution or the monotonicity of the prime subsequence within the frequency-ordered list; rather, it identifies these as open problems requiring deeper control over the factorization statistics of $p_{m-1} + 1$ . The work serves as a bridge between computational number theory observations and classical results on primes in arithmetic progressions.