A generalization of the Erd\H{o}s-Sierpinski conjecture

This paper investigates the equation $\sigma(n+1) = k\sigma(n)$ by combining combinatorial generalizations of Zumkeller numbers with advanced probabilistic number theory techniques to prove that the solution set has zero natural density with an explicit upper bound of $O(x/\sqrt{\log \log \log x})$, while also establishing conditional infinitude for the case $k=2$ under Schinzel's H Hypothesis.

The Gist

Imagine you are looking at a giant, infinite line of numbers, starting from 1 and going on forever: 1, 2, 3, 4, 5...

Every one of these numbers has a "family" of divisors (numbers that divide into it evenly). If you add up all the divisors of a number, you get a total called the sum of divisors. Let's call this sum $\sigma(n)$.

For example:

• The divisors of 6 are 1, 2, 3, and 6. Their sum is $1+2+3+6 = 12$.
• The divisors of 5 are just 1 and 5. Their sum is $1+5 = 6$.

The Big Question

Mathematicians have long been fascinated by a specific puzzle: How often does the sum of divisors of a number relate to the sum of divisors of the very next number?

The famous Erdős-Sierpiński conjecture asks if there are infinitely many times where the sum of divisors of a number is exactly the same as the sum of divisors of the next number (i.e., $\sigma(n+1) = \sigma(n)$). This is like asking, "How often do two neighbors have the exact same total weight?"

This paper takes that idea and makes it more general. Instead of asking if the sums are equal, it asks: How often is the sum of divisors of the next number exactly $k$ times bigger than the current one?
The equation is: $\sigma(n+1) = k \times \sigma(n)$.

Here, $k$ is any whole number bigger than 1 (like 2, 3, 4, etc.).

• If $k=2$, the next number's divisor sum is double the current one.
• If $k=3$, it's triple, and so on.

The Two Main Discoveries

The author, Amirali Fatehizadeh, tackles this problem from two different angles, using a mix of "counting" logic and "probability" logic.

1. The "Rarity" Discovery (The Probability Part)

The first major goal was to figure out how common these special numbers are. Do they appear frequently, or are they rare gems?

To answer this, the author used a clever trick from probabilistic number theory. Imagine trying to predict the weather. You can't predict the exact temperature for every single day forever, but you can model the probability of rain.

The author treated the numbers like a game of chance. They imagined that the "divisor sums" of consecutive numbers behave somewhat like independent random events (like flipping coins), even though they are mathematically linked.

• The Analogy: Imagine you are trying to find two people standing next to each other in a crowd who have a very specific, rare combination of traits (like having a specific height, shoe size, and favorite color).
• The Result: The author proved that finding these specific "neighbors" is incredibly difficult. In fact, as you look at larger and larger groups of numbers, the percentage of numbers that satisfy this equation drops to zero.

Even though there might be thousands of these numbers, they are so sparse that if you picked a number at random from a huge list, the chance of it being one of these special numbers is effectively zero. The paper provides a specific formula showing just how slowly they appear, proving they are "asymptotically rare."

2. The "Existence" Discovery (The Construction Part)

If these numbers are so rare, do they even exist? And are there infinitely many of them?

• For $k=2$: The author found a specific recipe (using polynomials) to generate these numbers. By assuming a famous mathematical hypothesis (Schinzel's H Hypothesis), they proved that there are infinitely many solutions where the next number's divisor sum is exactly double the current one.
• The General Guess: Based on the patterns found for $k=2$ and computer searches for $k=3$, the author proposes a bold guess: For any whole number $k$, there are infinitely many solutions.

Connecting to "Layered" Numbers

The paper also connects this to a fun combinatorial concept called k-layered numbers.

• The Analogy: Imagine you have a pile of bricks (the divisors of a number). Can you split these bricks into $k$ separate piles, where every single pile weighs exactly the same?
• If you can do this, the number is called "k-layered."
• The paper shows that the numbers satisfying our equation ($\sigma(n+1) = k\sigma(n)$) are deeply connected to these "layered" numbers. In fact, the solutions often have the perfect structure to be split into equal layers, avoiding the category of "weird numbers" (numbers that are abundant but can't be split evenly).

Summary in Plain English

1. The Puzzle: We are looking for pairs of consecutive numbers where the second one's "divisor sum" is exactly $k$ times the first one's.
2. The Density: These pairs are extremely rare. If you look at a huge range of numbers, the fraction of them that fit this rule is zero. They are like finding a specific grain of sand on a beach that keeps getting bigger.
3. The Infinity: Despite being rare, they likely never stop appearing. For the case where the ratio is 2 ($k=2$), the author proved (conditionally) that there are infinitely many of them.
4. The Structure: These special numbers have a very organized internal structure, allowing their divisors to be split into equal groups, much like a perfectly balanced scale.

In short, the paper proves that while these mathematical "miracles" are vanishingly rare in the grand scheme of numbers, they are not a fluke—they happen infinitely often and follow a beautiful, structured pattern.

Technical Summary

Technical Summary: A Generalization of the Erdős-Sierpiński Conjecture

Problem Statement
This paper investigates the combinatorial structure and asymptotic distribution of the solution set for the equation $\sigma(n+1) = k\sigma(n)$, where $k > 1$ is a fixed integer and $\sigma(n)$ denotes the sum-of-divisors function. This equation represents a generalization of the famous Erdős-Sierpiński conjecture (the case $k=1$), which posits the existence of infinitely many solutions to $\sigma(n+1) = \sigma(n)$. The primary objectives are to determine the natural density of the solution set $A_k = \{n \in \mathbb{N} : \sigma(n+1) = k\sigma(n)\}$ and to derive explicit quantitative upper bounds for its counting function $A_k(x) = \#\{n \le x : n \in A_k\}$. Additionally, the paper explores the existential nature of these solutions, particularly for the case $k=2$, linking them to the concepts of $k$-layered and Zumkeller numbers.

Methodology
The research employs a synthesis of combinatorial number theory and probabilistic number theory. The core analytical strategy involves the following steps:

1. Logarithmic Transformation and Additive Functions: The main equation is transformed via logarithms into an additive form involving the function $g(n) = \log(\sigma(n)/n)$. The problem is rephrased as investigating the probability that the difference $g(n+1) - g(n)$ falls within a small neighborhood of $\log k$.
2. Truncation and the Kubilius Model: To handle the quasi-independence of divisibility events, the authors utilize the Kubilius model. The additive function $g(n)$ is truncated to depend only on prime factors up to a bound $y$. By applying the Chinese Remainder Theorem, the arithmetic distribution of the truncated function over the integers is approximated by a synthetic measure space equipped with independent random variables.
3. Anti-Concentration Inequalities: Unlike standard applications of the Kubilius model which often seek central limit theorems for global distributions, this paper focuses on local behavior. The authors employ the Lévy concentration function and an optimized version of the Kolmogorov-Rogozin anti-concentration inequality (specifically Petrov's theorem). This allows for the bounding of the probability that the sum of independent random variables concentrates around a specific target value.
4. Error Control: The proof rigorously bounds the error introduced by truncation, the approximation of the arithmetic space by the probabilistic model, and the contribution of "large" prime factors or high prime powers. These error sets are shown to be negligible compared to the main probabilistic term.
5. Existential Analysis: For the specific case of $k=2$, the paper utilizes Schinzel's H Hypothesis. By constructing specific polynomial families, the authors demonstrate that if the hypothesis holds, there exist infinitely many integers $n$ satisfying the equation.

Key Contributions and Results

• Main Theorem (Asymptotic Density): The paper establishes that for any integer $k > 1$, the natural density of the solution set $A_k$ is zero. Specifically, the counting function satisfies the explicit upper bound:
  $$A_k(x) \ll_k \frac{x}{\sqrt{\log \log \log x}}$$
  This result implies that while solutions may exist, they are asymptotically rare among the natural numbers.
• Conditional Infinitude: Under the assumption of Schinzel's H Hypothesis, the paper proves that the equation $\sigma(n+1) = 2\sigma(n)$ possesses infinitely many solutions. This is demonstrated by constructing a polynomial family where the simultaneous primality of certain factors guarantees the equation holds.
• Computational Findings: The authors provide explicit lists of solutions for small values of $k$ (specifically $k=2$ and $k=3$) found via computational search, confirming that the solution sets are non-empty.
• Structural Connections: The paper elucidates the relationship between the solutions and $k$-layered numbers (a generalization of Zumkeller numbers). It observes that solutions often satisfy the necessary algebraic conditions for being $k$-layered (specifically $\sigma(N) \equiv 0 \pmod k$ for $N=n+1$) and frequently exhibit properties of practical and abundant numbers.

Significance and Claims
The paper claims to bridge the gap between the combinatorial classification of integers (such as perfect, Zumkeller, and $k$-layered numbers) and the analytic study of the sum-of-divisors function on consecutive arguments.

The primary significance lies in the derivation of a quantitative bound for the solution set of $\sigma(n+1) = k\sigma(n)$, a problem previously studied only in specific existential or elementary contexts (e.g., Torabi and Fatehizadeh, 2020). By proving the zero density, the work confirms that the coincidence of scaled sum-of-divisors values on consecutive integers is a rare phenomenon, extending the known behavior of similar equations (such as those studied by Erdős, Pomerance, and Sárközy).

Furthermore, the paper formulates Conjecture 4.2, generalizing the Erdős-Sierpiński conjecture to all $k > 1$: "For any positive integer $k$, the equation $\sigma(n+1) = k\sigma(n)$ has infinitely many solutions." While the paper only proves the conditional infinitude for $k=2$, it posits this as a natural extension based on the structural connections observed.

The work does not claim to resolve the Erdős-Sierpiński conjecture itself (the $k=1$ case) nor does it prove the unconditional infinitude of solutions for $k>1$. Instead, it provides a robust analytic framework for bounding the density of solutions and establishes a conditional existence result that motivates further investigation into the distribution of these special integers.