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Calculating the natural density of Mersenne numbers using nonstandard mathematical analysis

Using nonstandard mathematical analysis and a two-dimensional matrix based on the Pepis-Kalmar pairing function, the paper proposes that the natural density of Mersenne numbers is an infinitesimal value equivalent to the reciprocal of the infinite sum of the reciprocals of odd numbers.

Original authors: Gennady Eremin

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gennady Eremin

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 Question: How "Crowded" are Mersenne Numbers?

Imagine you have an infinite line of people numbered 0, 1, 2, 3, and so on. This is the set of all natural numbers. Now, pick out a specific group of people from this line: the Mersenne numbers (0, 1, 3, 7, 15, 31...). These are numbers that are one less than a power of two.

Mathematicians have long wondered: If you look at a huge chunk of this line, what percentage of the people are Mersenne numbers?

Is it 50%? (Like even numbers).
Is it 10%?
Or is it so tiny that it's effectively zero?

For a long time, this was an unsolved mystery. This paper claims to solve it using a special kind of math called Nonstandard Analysis.

The Tool: A Giant Spreadsheet (The Matrix)

To solve this, the author doesn't just look at a single line of numbers. He builds a giant, two-dimensional spreadsheet (a matrix) that contains every single natural number exactly once. No duplicates, no missing numbers.

He uses a clever sorting rule (the Pepis-Kalmar pairing function) to organize this spreadsheet:

The Rows (Horizontal): These are simple arithmetic patterns. If you look at any single row, the numbers are evenly spaced.
The Columns (Vertical): These are the interesting part. Each column starts with an even number (the "root") and then follows a specific rule to generate the next number: Multiply by 2 and add 1.
- Example: Start with 0. Next is $2(0)+1 = 1$ . Next is $2(1)+1 = 3$ . Next is $2(3)+1 = 7$ .
- This creates a "tree" of numbers growing upward. The very first column is the Mersenne numbers. The next column is a similar family called Thabit numbers, and so on.

The "Nonstandard" Twist: Infinite and Infinitesimal

Standard math usually deals with limits (approaching zero). This paper uses Nonstandard Analysis, which treats "infinitely small" and "infinitely large" numbers as actual, fixed values.

Think of it like this:

Infinitely Large ( $\omega$ ): Imagine a number so big it's bigger than any number you can count to.
Infinitesimal ( $\epsilon$ ): Imagine a number so small it's smaller than any fraction you can write, but it is not zero. It's a "ghost" of a number.

The Calculation: Balancing the Equation

The author looks at the "density" (how crowded the numbers are) in the columns of his spreadsheet.

The Rows are Easy: If you look at the rows, the math is simple. The first row has a density of 1/2, the second 1/4, the third 1/8, and so on. If you add all these up ( $1/2 + 1/4 + 1/8...$ ), you get exactly 1. This makes sense because the whole spreadsheet represents 100% of all numbers.
The Columns are Tricky:
- The first column (Mersenne numbers) is so sparse that its density is an infinitesimal number. Let's call this tiny amount $\epsilon$ .
- The second column (Thabit numbers) is even sparser. Its density is $\epsilon / 3$ .
- The third column is even sparser: $\epsilon / 5$ .
- The fourth: $\epsilon / 7$ .
- And so on, forever.

The author argues that if you add up the density of every single column, you must get back to 1 (because the columns cover all natural numbers).

So, he sets up this equation:
$1 = \epsilon + \frac{\epsilon}{3} + \frac{\epsilon}{5} + \frac{\epsilon}{7} + \dots$

If you factor out the tiny $\epsilon$ , it looks like this:
$1 = \epsilon \times \left( 1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \dots \right)$

The Conclusion

The part in the parentheses is the sum of the reciprocals of all odd numbers ( $1 + 1/3 + 1/5...$ ). In standard math, this sum goes to infinity. In this paper's nonstandard view, this sum is an infinitely large number, which the author calls $\omega$ .

So the equation becomes:
$1 = \epsilon \times \omega$

To find the density of Mersenne numbers ( $\epsilon$ ), you just divide 1 by that huge number:
$\epsilon = \frac{1}{\omega}$

What This Means

The paper claims two main things:

The density is NOT zero. Because $\epsilon$ is a specific, non-zero infinitesimal number, Mersenne numbers do exist in the set of natural numbers with a "positive" (though incredibly tiny) density.
The exact value. The density is exactly the reciprocal of the sum of the reciprocals of all odd numbers.

In simple terms: The author built a giant grid to organize all numbers. He found that while Mersenne numbers are incredibly rare (so rare their density is an "infinitesimal"), they are not non-existent. Their "rarity" is mathematically balanced against the infinite sum of odd fractions to equal exactly 1.

Note: The paper presents this as a theoretical solution using specific nonstandard mathematical tools. It does not discuss practical applications, such as cryptography or computer science uses, but focuses purely on solving this theoretical number theory puzzle.

Technical Summary: Calculating the Natural Density of Mersenne Numbers Using Nonstandard Mathematical Analysis

Problem Statement
The paper addresses a longstanding open problem in number theory: determining the natural density of the sequence of Mersenne numbers within the set of non-negative integers ( $\mathbb{N} = \{0, 1, 2, \dots\}$ ). Specifically, the author investigates whether this density is zero or a positive value. While standard analysis suggests the density tends toward zero as the range increases, the paper seeks to resolve the nature of this limit using nonstandard mathematical analysis.

Methodology
The author employs a two-pronged approach combining combinatorial number theory and nonstandard analysis:

Matrix Construction via Pepis-Kalmar Pairing:
The author constructs a two-dimensional matrix $\mathbb{M}$ containing all non-negative integers without repetition. This matrix is generated using the Pepis-Kalmar pairing function:
$f(x, y) = (2x + 1) \times 2^y - 1$
- Structure: The $x$ -axis represents even numbers (starting from 0), and the $y$ -axis represents the depth of the sequence.
- Columns (Mersenne Trees): Each column corresponds to a specific $x$ $x$ value and forms a "unary tree" (or Mersenne tree) where terms are generated by the recurrence $t' = 2t + 1$ $t^{'} = 2 t + 1$ .
  - The initial column ( $x=0$ ) contains the Mersenne numbers ( $2^y - 1$ ).
  - The second column ( $x=1$ ) contains Thabit numbers ( $3 \times 2^y - 1$ ).
  - Subsequent columns follow the pattern $(2x+1) \times 2^y - 1$ .
- Uniqueness: The author proves that all terms in this matrix are unique, ensuring that the matrix partitions the set of natural numbers $\mathbb{N}$ .
Nonstandard Analysis Framework:
The paper utilizes nonstandard analysis, which introduces actual infinitesimal ( $\epsilon$ ) and infinitely large ( $\omega$ ) quantities as constant numbers rather than variables.
- Rows: The natural density of each row (an arithmetic progression) is calculated as the inverse of its common difference ( $1/2, 1/4, 1/8, \dots$ ). The sum of these densities equals 1, consistent with the density of $\mathbb{N}$ .
- Columns: The author analyzes the density of the columns. For the initial column (Mersenne numbers), the density is denoted as an infinitesimal $\epsilon$ . For subsequent columns ( $x=1, 2, \dots$ ), the densities are derived as $\epsilon/3, \epsilon/5, \dots$ , corresponding to the odd factors $(2x+1)$ .

Key Contributions and Results
The primary contribution is the derivation of a specific equation relating the natural density of Mersenne numbers to the sum of the reciprocals of odd numbers.

Density Derivation: By summing the densities of all columns in the matrix, the author equates the total density to 1 (the density of $\mathbb{N}$ ). This yields the equation:
$1 = \epsilon + \frac{\epsilon}{3} + \frac{\epsilon}{5} + \frac{\epsilon}{7} + \dots = \epsilon \times \left(1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \dots\right)$
Definition of $\omega$ : The term in the parentheses is identified as $\omega$ , the sum of the reciprocals of all odd natural numbers, which is an infinitely large value in nonstandard analysis.
The Main Equation: The relationship is formalized as:
$\epsilon \times \omega = 1 \quad \text{or} \quad \epsilon = \frac{1}{\omega}$
Theorem 2: The paper concludes that the natural density of Mersenne numbers ( $\epsilon$ ) and the reciprocal of the sum of reciprocals of odd natural numbers ( $1/\omega$ ) are equivalent infinitesimal numbers.

Significance and Claims
The paper claims to solve the open problem regarding the natural density of Mersenne numbers by proving that the density is positive (non-zero).

Proof of Positivity: The author argues that if $\epsilon$ were zero, the sum of the column densities would be zero, contradicting the fact that the matrix covers all natural numbers (density = 1). Therefore, $\epsilon$ must be a positive infinitesimal.
Concrete Determination: While acknowledging that the distribution density of Mersenne numbers is traditionally unknown in standard mathematics, the paper asserts that it has obtained a "concrete equation" for determining this density within the framework of nonstandard analysis.
Equivalence: The result establishes an equivalence between the density of Mersenne numbers and the reciprocal of the divergent harmonic series of odd numbers, treating both as specific, non-zero infinitesimal values within the nonstandard number system.

The paper does not propose experimental applications or future extensions beyond the mathematical derivation presented. The significance is framed strictly within the context of resolving the theoretical question of Mersenne number density using the tools of nonstandard analysis.