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Solution of Erd\H{o}s problem $\# 443$

This paper resolves Erdős problem #443 by proving that the size of the intersection between the sets of products $\{k(m-k)\}$ and $\{l(n-l)\}$ is bounded by $(mn)^{o(1)}$ yet can still be arbitrarily large.

Original authors: Stijn Cambie

Published 2026-07-29

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Stijn Cambie

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 a world where numbers aren't just cold, hard digits, but rather players in a giant, invisible game of hide-and-seek. This is the realm of number theory, a branch of mathematics that treats integers like unique characters with secret identities. In this game, we often look at "sets"—which are just fancy words for collections of numbers—created by following a specific rule. For instance, if you take a number, multiply it by its partner (the number that adds up to a certain total), and list all the results, you get a unique pattern. Mathematicians love asking: "If I make two different patterns using different rules, how many numbers will they have in common?" It's like asking how many words appear in both a dictionary of ancient poetry and a dictionary of modern slang. The question might seem like a puzzle for a math club, but it helps us understand the hidden architecture of numbers, revealing whether patterns are rare, common, or completely unpredictable.

The paper you're about to hear about tackles a specific riddle posed by the legendary mathematician Paul Erdős. He wondered about two special collections of numbers. The first collection is made by taking a number $m$ , picking a smaller number $k$ (from 1 up to half of $m$ ), and calculating the product $k(m-k)$ . The second collection does the exact same thing but with a different number, $n$ . The big question was: As these numbers get huge, how many "common friends" (numbers that appear in both lists) can they share? Erdős guessed that while the number of shared friends might grow, it would grow very slowly—so slowly that for any tiny margin of error you pick, the count would eventually be smaller than a specific mathematical formula involving the size of the numbers. He also asked if this number of shared friends could grow without ever stopping, or if it would hit a ceiling.

The author of this paper, Stijn Cambie, acts like a detective solving this decades-old mystery. He confirms that the number of shared friends is indeed unbounded, meaning it can get as large as you want if you pick the right numbers $m$ and $n$ . To prove this, he uses a clever trick: he shows that finding a shared number is the same as finding a way to break a specific difference of squares into two smaller pieces. This turns the problem into counting the "divisors" (the building blocks) of a number. Since we know that some numbers have an enormous number of divisors, Cambie proves that we can always find pairs of $m$ and $n$ that create a massive number of shared friends.

However, the paper also puts a strict speed limit on this growth. Cambie demonstrates that while the number of shared friends can get huge, it grows incredibly slowly—so slow that it fits the "tiny margin" guess Erdős made. He shows that the count is bounded by a function that is essentially "almost constant" compared to the size of the numbers involved. In plain terms, even if you pick the best possible numbers to maximize the overlap, the number of shared friends will never explode; it will always remain a tiny fraction of the total numbers involved.

Interestingly, the paper reveals a twist in the tale: this problem wasn't actually a new discovery. The author notes that a mathematician named Norbert Hegyvári solved this exact problem 40 years earlier, but his proof was only recently published. So, while this paper provides a fresh, clear explanation and confirms the answer, the "solved" status of the problem actually belongs to that earlier, long-hidden work. The paper doesn't just guess; it provides a mathematical proof, showing exactly how the number of shared friends behaves and confirming that it is both unbounded and surprisingly small relative to the size of the numbers used.

Technical Summary: Solution of Erdős Problem #443

Problem Statement
This paper addresses Problem #443 posed by Erdős and Graham in their 1980 work [1]. The problem concerns two sets of integers generated by the quadratic forms $k(m-k)$ and $\ell(n-\ell)$ , where $1 \le k \le m/2$ and $1 \le \ell \le n/2$ . Specifically, let $S(m) = \{k(m-k) : 1 \le k \le m/2\}$ and $S(n) = \{\ell(n-\ell) : 1 \le \ell \le n/2\}$ . The central questions are:

Can the number of common integers, denoted $f(n, m) = \#(S(m) \cap S(n))$ , be estimated?
Is this number unbounded as $m$ and $n$ vary?
Is the number of common elements strictly less than $(mn)^\epsilon$ for every $\epsilon > 0$ when $mn$ is sufficiently large?

Methodology and Derivation
The author, Stijn Cambie, adopts the notation $f(n, m)$ and assumes without loss of generality that $m > n$ . The core of the methodology involves transforming the intersection condition into a difference of squares.

By observing that $k(2m-k) = m^2 - (m-k)^2$ , the author sets $c = m-k$ . The condition for an integer to be in both sets, $k(2m-k) = \ell(2n-\ell)$ , is rewritten as:
$m^2 - c^2 = n^2 - d^2$
where $d = n-\ell$ . Rearranging terms yields:
$m^2 - n^2 = c^2 - d^2 = (c-d)(c+d)$
This transformation demonstrates that the number of solutions is bounded by half the number of divisors of the integer $m^2 - n^2$ , denoted as $\tau(m^2 - n^2)$ . The constraint that $c-d$ must be a divisor bounded by $\sqrt{m^2 - n^2}$ is utilized to establish this bound.

Key Results

Upper Bound: The paper establishes that $f(n, m) \le \lfloor \tau(m^2 - n^2) / 2 \rfloor$ . By invoking the well-known result regarding the divisor function $\tau(x)$ , which states that $\tau(x) \le 2^{(1+o(1))\frac{\log x}{\log \log x}} = x^{o(1)}$ , the author confirms that $f(n, m) = (mn)^{o(1)}$ . This affirms the conjecture that the number of common integers is less than $(mn)^\epsilon$ for any $\epsilon > 0$ when $mn$ is sufficiently large.
Unboundedness: To address whether the number of common integers is unbounded, the author considers the specific case where $m = n+1$ . In this scenario, $m^2 - n^2 = 2n+1$ . The number of solutions is shown to be related to $\lfloor \tau(2n+1)/2 \rfloor - 1$ . Since $\tau(x)$ is unbounded (achieving large values for highly composite numbers), $f(2n+2, 2n)$ is unbounded. The construction explicitly maps factorizations of $2n+1 = pq$ (with $p \ge q \ge 2$ ) to valid integer solutions for $c$ and $d$ .

Significance and Claims
The paper claims to provide a complete solution to Erdős Problem #443, confirming both the sub-polynomial growth rate of the intersection size and its unbounded nature.

The author includes a specific remark regarding the history of the solution: Norbert Hegyvári [2] solved this problem independently 40 years prior to the publication of this note. However, Hegyvári's proof was only published recently, after the submission of this note. The paper positions itself as an independent verification and solution to the problem, while acknowledging the prior, delayed work of Hegyvári.

References Cited in the Paper

[1] P. Erdős and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, 1980.
[2] N. Hegyvári, (Solution to the problem, published recently).
[3] Standard result on the divisor function $\tau(x)$ .

Technical Summary: Solution of Erdős Problem #443

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