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On the Asymptotic Density of a GCD-based Map

This paper establishes that the symmetry of the GCD-based map $f(a,b)$ arises from an $\operatorname{SL}_2(\mathbb{Z})$ action on primitive pairs, provides a uniform three-parameter description for its level sets, and determines the asymptotic densities of pairs satisfying $f(a,b)=1$ and its higher-order analogues.

Original authors: Thang Pang Ern, Malcolm Tan Jun Xi

Published 2026-04-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Thang Pang Ern, Malcolm Tan Jun Xi

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, infinite grid of numbers, like a chessboard that stretches forever in every direction. On every square of this board, you place two numbers: one for the row (let's call it A) and one for the column (let's call it B).

Now, imagine a magical machine that looks at every pair of numbers $(A, B)$ and spits out a single result based on a specific recipe:

Multiply them: $A \times B$
Add them: $A + B$
Find the "Greatest Common Divisor" (GCD) of those two results. (The GCD is the biggest number that divides both of them evenly).
Divide that result by the GCD of the original numbers $A$ and $B$ .

The paper asks a simple question: What happens if we look at all the numbers this machine produces?

1. The Magic Machine is Surjective (It Can Make Anything)

The authors discovered that this machine is incredibly versatile. No matter what whole number you pick (1, 2, 100, or a million), you can find a pair of numbers $(A, B)$ that makes the machine spit out exactly that number.

The Analogy: Think of this machine like a universal translator. No matter what "language" (number) you want to speak, there is a specific combination of inputs that will translate to it. The paper even gives you a "cheat sheet" (a formula) to find the right ingredients for any number you want.

2. The Hidden Symmetry (The Dance of the Numbers)

When the authors plotted the results on a grid (a "heat map"), they saw a beautiful, symmetric pattern. The top-left corner looked like a mirror image of the bottom-right.

The Analogy: Imagine a dance floor where pairs of dancers $(A, B)$ are moving. The paper reveals that these dancers aren't just moving randomly; they are following a strict, ancient choreography defined by a group of mathematicians called $SL_2(\mathbb{Z})$ .

Think of this group as a set of "magic moves." If you take a pair of numbers and apply a "magic move," you get a new pair.
Amazingly, even though the numbers change, the result of the machine stays the same.
The symmetry in the heat map is just the visual proof that these dancers are following the same rules, just mirrored.

3. The "1" Club (How Often Do We Get the Number 1?)

The most interesting part of the paper is asking: If we pick two random numbers, how often does the machine spit out the number 1?

In the world of numbers, getting a "1" is like finding a perfect match where everything cancels out nicely.

The authors calculated that if you look at a huge chunk of the grid (say, 70,000 by 70,000), about 88.15% of the pairs result in the number 1.
This number isn't random. It's a famous constant in mathematics known as the Quadratic Class Number Constant.

The Deep Connection:
Here is the mind-blowing part: This constant (0.88151) usually appears in a completely different branch of math called Algebraic Number Theory, where mathematicians study the "shape" of number systems (specifically, real quadratic fields).

The Metaphor: It's as if you were studying the weather patterns in a rainforest (our grid of numbers) and discovered that the frequency of rain perfectly matched the frequency of a specific type of crystal formation in a cave miles away. It suggests a hidden, deep connection between how numbers add/multiply and the fundamental structure of number systems.

4. The "Boring" Case (When the Power Changes)

The authors also asked: "What if we change the recipe slightly?"
Instead of adding $A+B$ , what if we added $A^2 + B^2$ or $A^3 + B^3$ ?

The Analogy:

When the power is 1 (our original recipe), the machine is very picky and produces a "1" about 88% of the time.
But when the power is 2 or higher, the machine becomes much more "relaxed." The result is almost always just the GCD of the original numbers.
In this case, the chance of getting a "1" drops to about 60.79% ( $6/\pi^2$ ).
This 60.79% is a very famous number in math: it's the probability that two random numbers are relatively prime (they share no common factors other than 1). It's like flipping a coin and getting heads, but for numbers.

Summary

This paper is a journey through a mathematical landscape:

Discovery: They found a function that can generate any number you want.
Structure: They found that the numbers follow a hidden, symmetrical dance (group theory).
Probability: They calculated that the number "1" appears about 88% of the time, a number that links the simple act of adding/multiplying to the complex structure of number fields.
Contrast: They showed that changing the rules slightly (raising numbers to a power) makes the pattern much simpler and more predictable.

In short, the paper shows that even in the simple act of adding and multiplying two numbers, there is a profound, hidden order that connects different worlds of mathematics.

1. Problem Statement

The paper investigates the arithmetic properties and asymptotic behavior of the two-variable function $f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ defined as:
$f(a, b) = \frac{\gcd(ab, a+b)}{\gcd(a, b)}$
The authors aim to address three primary questions:

Surjectivity: Does every natural number appear as an output of $f(a, b)$ ?
Characterization: What is the complete set of pairs $(a, b)$ that map to a specific integer $n$ ?
Density: What is the asymptotic density of pairs $(a, b)$ such that $f(a, b) = 1$ , and how does this generalize for $f_r(a, b)$ where $r \geq 2$ ?

2. Methodology

The authors employ a combination of elementary number theory, group theory, and analytic number theory:

Reduction to Primitive Pairs: The function is analyzed by decomposing $a$ and $b$ into $a = d a'$ and $b = d b'$ , where $d = \gcd(a, b)$ and $\gcd(a', b') = 1$ . This simplifies the function to $f(a, b) = \gcd(d, a' + b')$ .
Group Action ( $SL_2(\mathbb{Z})$ ): The authors analyze the symmetries of the solution space by considering the action of the modular group $SL_2(\mathbb{Z})$ on primitive pairs $(a', b')$ . They identify a stabilizer subgroup that preserves the sum $a' + b'$ , linking the geometric symmetry observed in heat maps to algebraic group orbits.
Chinese Remainder Theorem (CRT): For squarefree $n$ , the authors construct solutions by combining arithmetic progressions derived from prime factors of $n$ .
Probabilistic Modeling: To determine asymptotic density, the authors model the divisibility conditions of $d$ and $a'+b'$ by prime numbers as independent events, utilizing Euler products.
Analytic Tools: The paper utilizes the Dirichlet hyperbola method and properties of multiplicative functions (Euler totient $\phi$ and divisor sum $\sigma$ ) to derive limiting densities.

3. Key Contributions and Results

A. Surjectivity and Integer Property

Proposition 1.1: Proves that $f(a, b)$ is always an integer for all $a, b \in \mathbb{N}$ using Bézout's lemma.
Proposition 1.2: Proves that $f$ is surjective. For any $c \in \mathbb{N}$ , the pair $(c, c^2 - c)$ yields $f(c, c^2 - c) = c$ .

B. Structural Characterization of Solutions

Theorem 2.3 (General Parameterization): The authors provide a uniform three-parameter description for all solutions to $f(a, b) = n$ $f (a, b) = n$ . A pair $(a, b)$ $(a, b)$ satisfies $f(a, b) = n$ $f (a, b) = n$ if and only if:
$a = n k a', \quad b = n k b'$
where:
1. $m$ is a positive integer such that $a' + b' = nm$.
2. $a'$ is an integer with $1 \leq a' \leq nm - 1$ and $\gcd(a', nm) = 1$ .
3. $k$ is a positive integer coprime to $m$ ( $\gcd(k, m) = 1$ ).
Squarefree Case (Theorem 2.1): For squarefree $n = p_1 \dots p_k$ , the solutions form $2^k$ families of arithmetic progressions. These are constructed using the Chinese Remainder Theorem based on minimal pairs and gaps associated with each prime factor.

C. Asymptotic Density for $f(a, b) = 1$

Theorem 3.2: The limiting density $\rho$ of pairs $(a, b)$ such that $f(a, b) = 1$ is given by the Euler product:
$\rho = \prod_{p} \left( 1 - \frac{1}{p^2(p+1)} \right) \approx 0.88151$
Significance: This constant is identified as the quadratic class number constant, typically associated with the distribution of class numbers in real quadratic fields (Cohen-Lenstra heuristics). The paper establishes a novel connection between this additive number theory problem and algebraic number theory.
Analytic Interpretation: Lemma 3.3 shows this density is the asymptotic mean of the function $\frac{\phi(n)\sigma(n)}{n^2}$ .

D. Generalization to $f_r(a, b)$ for $r \geq 2$

Theorem 4.2: For the generalized function $f_r(a, b) = \frac{\gcd(a^r + b^r, ab)}{\gcd(a, b)}$ with $r \geq 2$ , the authors prove that $\gcd(a^r + b^r, ab) = \gcd(a, b)$ .
Result: Consequently, $f_r(a, b) = 1$ for all coprime pairs. The limiting density of pairs where $f_r(a, b) = 1$ is simply the density of coprime pairs:
$\rho_r = \frac{6}{\pi^2} \approx 0.60793$
This aligns with the classic result that the probability of two random integers being coprime is $6/\pi^2$ .

4. Significance

Bridging Fields: The paper highlights a profound, previously unexplored link between the density of a specific GCD-based map and the quadratic class number constant, a central object in the study of class groups of real quadratic fields.
Symmetry Analysis: It provides a rigorous group-theoretic explanation (via $SL_2(\mathbb{Z})$ ) for the visual symmetries observed in the function's heat map, moving beyond empirical observation to algebraic proof.
Uniform Solution Set: The derivation of a complete, three-parameter characterization for the pre-image of any integer $n$ offers a comprehensive structural understanding of the function, recovering known arithmetic progression families as special cases.
Methodological Demonstration: The work effectively demonstrates the utility of combining elementary number theory (CRT, Bézout) with analytic tools (Euler products, Dirichlet hyperbola method) to solve density problems.

5. Conclusion

The paper successfully characterizes the function $f(a, b)$ , proving its surjectivity and providing a complete parameterization of its level sets. Most notably, it calculates the asymptotic density of the level set $f(a, b)=1$ to be approximately $0.88151$, identifying this value as the quadratic class number constant. This result unifies concepts from additive number theory, group actions, and algebraic number theory, while showing that the behavior stabilizes to the standard coprime density ( $6/\pi^2$ ) for higher powers ( $r \geq 2$ ).