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On the Limiting Density of a gcd Map

This paper investigates the limiting density of the function $f(a,b)=\frac{\gcd(a+b,ab)}{\gcd(a,b)}$ , proving that the probability of $f(a,b)=1$ is approximately $0.88151$, a value given by an Euler product that coincidentally matches the quadratic class number constant from real quadratic fields, while also showing that its higher-order analogues yield the density $1/\zeta(2)$ .

Original authors: Thang Pang Ern, Malcolm Tan Jun Xi, Loh Wei Xuan Ryan

Published 2026-04-21

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Thang Pang Ern, Malcolm Tan Jun Xi, Loh Wei Xuan Ryan

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 two numbers, let's call them Alice ( $a$ ) and Bob ( $b$ ).

In the world of math, these two numbers have two main ways of interacting:

The Addition Party: They join forces to create a sum ( $a + b$ ).
The Multiplication Party: They multiply to create a product ($ab$).

Usually, these two parties are quite different. But sometimes, they share a secret: a common divisor. A common divisor is a number that divides both the sum and the product evenly.

The authors of this paper are playing a game with Alice and Bob. They ask: "How often do Alice and Bob have a sum and a product that share no common divisors other than 1?"

To measure this, they created a special "compatibility score" called $f(a, b)$ .

If the score is 1, it means Alice and Bob are perfectly "coprime" in this specific way (their sum and product don't share any hidden factors).
If the score is 2, 3, 4, etc., it means they share a common factor, and the score tells you how big that factor is.

The Big Discovery: The "88%" Rule

The researchers looked at millions of pairs of numbers (from 1 to 70,000 and beyond) to see how often the score was exactly 1.

They found a fascinating pattern. As they looked at larger and larger groups of numbers, the percentage of "perfectly compatible" pairs settled down to a very specific number:
Approximately 88.15%

In math terms, they call this the limiting density. It's like flipping a coin billions of times; eventually, you know the coin will land on heads 50% of the time. Here, if you pick two random numbers, there is an 88.15% chance that their "sum-product compatibility score" will be 1.

The "Magic Coincidence"

Here is where it gets really weird and beautiful.

This number, 0.88151, isn't just a random statistic from this specific game. It appears in a completely different branch of mathematics called Algebraic Number Theory, specifically when studying "Real Quadratic Fields" (a fancy way of describing certain types of number systems involving square roots).

In that field, this number is known as the Quadratic Class Number Constant. It helps mathematicians understand how "messy" or "broken" the rules of multiplication are in those specific number systems.

The Analogy:
Imagine you are studying the weather in a small town (our $f(a,b)$ game). You calculate that it rains 88% of the time. Then, you discover that this exact same percentage is the key to predicting the migration patterns of a rare bird species on a different continent (the Quadratic Fields).

The paper shows that the "weather" of our simple addition/multiplication game is mathematically identical to the "migration patterns" of these complex number systems. It's a hidden bridge between two worlds that look nothing alike.

What Happens if We Change the Game?

The authors asked: "What if we change the rules?"

Instead of adding ( $a + b$ ), what if we used higher powers, like cubes ( $a^3 + b^3$ ) or fourth powers ( $a^4 + b^4$ )?

They found that as soon as you go beyond simple addition (using powers of 2 or higher), the "magic" disappears. The compatibility score stops being a complex puzzle and collapses into a very simple, well-known rule: Are the two numbers coprime?

If you just ask, "Do $a$ and $b$ share any factors?" the answer is 60.79% (which is $6/\pi^2$ ).

So, the "88%" magic only happens when you mix Addition and Multiplication in that specific, delicate way. As soon as you make the addition "stronger" (by using higher powers), the subtle interaction vanishes, and you are left with the standard rules of the game.

Summary for the Everyday Reader

The Game: Take two numbers, add them, multiply them, and see if the results share any common factors.
The Result: About 88.15% of the time, they don't share any factors.
The Surprise: This 88.15% number is the same as a famous constant used to study deep, complex structures in number theory. It's a hidden link between a simple arithmetic game and the fundamental nature of numbers.
The Lesson: Simple interactions between addition and multiplication can hide deep, universal truths that connect different areas of mathematics.

The paper is essentially a detective story where the authors found a fingerprint (the 88.15% number) at a simple crime scene (the $f(a,b)$ function) and realized it belonged to a master criminal in a completely different part of the city (Quadratic Fields).

1. Problem Statement

The paper investigates the behavior of a specific function $f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ defined as:
$f(a, b) = \frac{\gcd(a + b, ab)}{\gcd(a, b)}$
This function encapsulates the interaction between additive ( $a+b$ ) and multiplicative ($ab$) structures. The primary research question is to determine the limiting density $\rho$ of pairs $(a, b)$ such that $f(a, b) = 1$ . In other words, the authors seek the probability that the additive combination $a+b$ and the multiplicative combination $ab$ share no common divisors beyond those already forced by $\gcd(a, b)$ .

Additionally, the paper explores a higher-order analogue $f_r(a, b)$ where $a+b$ is replaced by $a^r + b^r$ for $r \geq 2$ , to determine if the unique behavior of the linear case persists.

2. Methodology

The authors employ a combination of elementary number theory, probabilistic methods over integers, and analytic number theory.

Surjectivity Analysis: The authors first establish that $f$ is a surjective map onto $\mathbb{N}$ , proving that every natural number can be attained as a value of $f(a, b)$ .
Probabilistic Decomposition: To find the density where $f(a, b) = 1$ , the authors decompose $a$ and $b$ as $a = d a'$ and $b = d b'$ , where $d = \gcd(a, b)$ and $\gcd(a', b') = 1$ . The condition $f(a, b) = 1$ is equivalent to $\gcd(a' + b', d) = 1$ .
Local-to-Global Principle (Euler Product): The authors analyze the probability $P_p$ that a prime $p$ divides both $d$ and $a' + b'$ . Using the Chinese Remainder Theorem, they treat events for distinct primes as independent, allowing the global density to be expressed as an infinite product over all primes:
$\rho = \prod_{p} (1 - P_p)$
Exact Divisibility Analysis: The calculation of $P_p$ involves analyzing the exact power of $p$ dividing $a$ and $b$ (denoted $p^k \parallel n$ ). The authors sum over all possible powers $k \geq 1$ to determine the probability that $p$ divides the reduced sum $a' + b'$ .
Analytic Number Theory Connections: The paper connects the resulting Euler product to known constants in algebraic number theory and utilizes the Dirichlet hyperbola method to analyze the asymptotic behavior of related arithmetic functions (Euler totient $\phi$ and sum of divisors $\sigma$ ).

3. Key Contributions and Results

A. The Limiting Density Constant

The main result is the derivation of the limiting density $\rho$ :
$\rho = \prod_{p} \left( 1 - \frac{1}{p^2(p+1)} \right) \approx 0.88151$
This value is derived by calculating the probability $P_p = \frac{1}{p^2(p+1)}$ that a prime $p$ divides both the GCD $d$ and the sum of the coprime parts $a' + b'$ .

B. Connection to Algebraic Number Theory

The authors identify a surprising and significant connection: the constant $\rho$ is identical to the quadratic class number constant appearing in the asymptotic analysis of class numbers for real quadratic fields $\mathbb{Q}(\sqrt{D})$ .

Specifically, it appears in the asymptotic formula for the sum of class numbers $h(D)$ weighted by the regulator $R(D)$ , as proven by Ayoub (1963).
This establishes a novel link between a combinatorial GCD map and the arithmetic invariants of quadratic fields.

C. Connection to Group Theory

The paper reveals that the quantity $\phi(n)\sigma(n)$ (where $\phi$ is Euler's totient and $\sigma$ is the sum of divisors) counts the number of conjugacy classes in the general linear group $GL(2, \mathbb{Z}/n\mathbb{Z})$ .

The map $(a, b) \mapsto (a+b, ab)$ corresponds to the trace and determinant of a companion matrix.
The density $\rho$ essentially measures the frequency with which randomly chosen integers $(a, b)$ generate a "primitive" pair of trace and determinant, analogous to the distribution of characteristic polynomials in $GL(2, \mathbb{Z}/n\mathbb{Z})$ .

D. Higher-Order Analogue ( $r \geq 2$ )

The authors define $f_r(a, b) = \frac{\gcd(a^r + b^r, ab)}{\gcd(a, b)}$ .

Result: For $r \geq 2$ , the condition $f_r(a, b) = 1$ collapses to the simple condition $\gcd(a, b) = 1$ .
Density: The limiting density becomes the classical probability of two integers being coprime:
$\rho_r = \frac{1}{\zeta(2)} = \frac{6}{\pi^2} \approx 0.6079$
Significance: This highlights that the unique interaction between addition and multiplication observed in the linear case ( $r=1$ ) is specific to the linear structure; higher powers decouple the additive and multiplicative constraints.

4. Significance

Interdisciplinary Link: The paper provides a concrete example of how a simple number-theoretic function can yield a constant that bridges combinatorial probability, analytic number theory (Euler products), and algebraic number theory (class numbers of quadratic fields).
Structural Insight: It demonstrates that the "interaction" between addition and multiplication is most profound in the linear case. The collapse of the density to $6/\pi^2$ for $r \geq 2$ suggests that higher-degree sums lose the specific divisibility constraints present in the linear sum $a+b$ .
Computational Verification: The theoretical density $\approx 0.88151$ is supported by empirical data (heat maps and large-scale simulations up to $n=70,000$ ), confirming the stability of the limit.

Conclusion

The paper successfully characterizes the density of pairs $(a, b)$ where the GCD map $f(a, b)$ equals 1. It derives an exact Euler product formula, identifies its value as the quadratic class number constant, and contextualizes the result within the theory of conjugacy classes in $GL(2, \mathbb{Z}/n\mathbb{Z})$ . The work underscores the unique arithmetic properties of the linear sum $a+b$ compared to higher-degree sums.