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A Complete Answer to Erd\H{o}s Problem 690

This paper resolves Erdős's Problem 690 by proving that the natural density of integers with a specific $k$ -th smallest prime divisor is not unimodal for any $k \ge 4$ , thereby completing the classification of this property for all $k$ .

Original authors: Shouqiao Wang, Davide Crapis

Published 2026-05-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shouqiao Wang, Davide Crapis

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 Picture: A Mystery About Number Patterns

Imagine you have a giant, infinite list of all the prime numbers (2, 3, 5, 7, 11, 13...). Now, imagine you are looking at every single whole number (1, 2, 3, 4...) and asking a specific question: "What is the k-th smallest prime number that divides this whole number?"

For example, if you pick the number 12, its prime factors are 2 and 3.

The 1st smallest prime factor is 2.
The 2nd smallest prime factor is 3.

The mathematician Paul Erdős wondered about a pattern in how often these "k-th smallest" primes appear. He asked: Does the frequency of these primes go up to a peak and then go down, like a smooth hill? In math terms, he asked if the sequence is "unimodal" (one single hump).

The Hill Analogy: Imagine a mountain. As you walk up, the view gets better (frequency goes up). You reach the very top (the peak). Then, as you walk down the other side, the view gets worse (frequency goes down). Erdős thought this "mountain shape" might be true for every level of the "k-th smallest" prime.

What Was Already Known?

Before this paper, a mathematician named Cambie had checked the first few levels:

For k=1, 2, and 3, the "mountain" shape was real. The frequency went up, peaked, and went down.
For k=4 up to 20, Cambie found that the shape was broken. It wasn't a smooth hill; it had weird bumps and dips.

But the big question remained: Does this broken pattern happen for every number k greater than 3? Or does the "mountain" eventually reappear for very large numbers?

The Discovery: The "Multiscalar Fields System"

The authors, Shouqiao Wang and Davide Crapis, didn't just sit at a desk and do this by hand. They built a digital assistant called the Multiscalar Fields System.

Think of this system as a super-smart, tireless explorer equipped with a map and a compass.

Exploration: The system looked at millions of possibilities, testing different mathematical arguments to see which ones held water.
Refinement: When an argument was weak, the system tweaked it. When it was strong, it kept it.
Verification: It checked its own math against strict rules to ensure no errors were made.

Human mathematicians acted as the "auditors." They set the goal, checked the final proof, and verified the computer's calculations, but the heavy lifting of discovery was done by the system.

The Solution: Finding the "Valleys" and "Peaks"

The paper proves that for every single k greater than 3, the "mountain" shape is false. The sequence never settles into a single smooth hill.

To prove this, the authors used a clever two-step strategy, like finding a specific path through a forest:

1. The "Descent" (Going Down the Hill)
They found a specific spot in the list of prime numbers where a huge gap appeared between two primes.

Analogy: Imagine walking up a hill, and suddenly you hit a massive, wide canyon. The path drops sharply.
The Math: They proved that when the gap between two primes is huge, the frequency of the "k-th smallest prime" drops sharply. This is the "descent."

2. The "Ascent" (Climbing Back Up)
Later in the list, they found a spot where the primes were very close together (a tiny gap).

Analogy: After the canyon, you find a steep, narrow staircase leading back up.
The Math: They proved that when the gap between primes is tiny, the frequency shoots back up. This is the "ascent."

The Conclusion:
If a path goes down (descent) and then later goes up (ascent), it cannot be a single smooth hill. It must have a "valley" in the middle. Therefore, the sequence is not unimodal.

How They Proved It for All Numbers

The paper splits the proof into two parts, like solving a puzzle with a small section and a huge section:

The Small Numbers (k = 4 to 8,600,001):
For these, the system used certified certificates. Think of these as "official receipts" from other mathematicians who had already found specific, record-breaking prime gaps (like a massive canyon) and twin primes (like a tiny staircase). The system plugged these known facts into their formulas to prove the "down-then-up" pattern existed for every number in this range.
The Huge Numbers (k = 8,600,002 and beyond):
For numbers this big, you can't just look up a receipt. You have to build the path yourself.
The authors used a Chinese Remainder Construction.
- Analogy: Imagine you want to build a long wall of bricks where every brick is "composite" (not prime). You use a special recipe (the Chinese Remainder Theorem) to arrange the bricks so that no matter where you look, there's always a prime factor hiding in the pattern.
- This allowed them to mathematically guarantee the existence of a massive canyon (a huge gap) followed later by a tiny staircase (a small gap), proving the pattern holds forever, no matter how big k gets.

The Final Verdict

The paper provides the complete answer to Erdős's question:

k = 1, 2, 3: The sequence is a smooth hill (Unimodal).
k ≥ 4: The sequence is a jagged, bumpy path with valleys and peaks (Not Unimodal).

The "Multiscalar Fields System" successfully navigated the complex landscape of prime numbers to show that for any level of "k-th smallest prime" beyond the third, the pattern of their frequency is never a simple, single hill. It is always a rollercoaster that goes down and then back up.

Technical Summary: A Complete Answer to Erdős Problem 690

Problem Statement
The paper addresses Erdős Problem 690, which concerns the natural density $d_k(p)$ of positive integers $n$ for which a prime $p$ is the $k$ -th smallest prime divisor. The central question, posed by Erdős, asks whether the sequence $p \mapsto d_k(p)$ , indexed by primes in increasing order, is unimodal for every fixed integer $k \ge 1$ . A sequence is unimodal if it is non-decreasing up to a certain index and non-increasing thereafter.

Prior work by Cambie established that the sequence is unimodal for $1 \le k \le 3$ and verified non-unimodality for the initial range $4 \le k \le 20$ . The open problem remained whether non-unimodality holds for all $k \ge 4$ .

Methodology and Framework
The authors developed the Multiscalar Fields System, an automated framework for large-scale mathematical exploration, theorem discovery, and verification. In this work, the system autonomously explored candidate constructions and arguments, evaluated their mathematical viability, and iteratively refined them to produce a formal proof. Human involvement was restricted to problem formulation, auditing the generated proof, and verifying numerical computations.

The proof strategy relies on two main components:

Structural Lemmas: The authors derive an exact threshold criterion (Lemma 3.1) determining the sign of the first difference $d_{r+1}(p_{i+1}) - d_{r+1}(p_i)$ . This criterion compares a ratio $R_r(i)$ of elementary symmetric functions of prime weights to the adjacent prime gap $g_i$ . Specifically, a strict descent occurs if $R_r(i-1) < g_i + 1$ , and a strict ascent occurs if $R_r(i-1) > g_i + 1$ .
Construction of Non-Unimodality: To prove non-unimodality for a given $k$ $k$ , the authors construct a scenario where the sequence exhibits a strict descent followed later by a strict ascent. This requires finding:
- An earlier prime gap large enough to force a descent.
- A later prime gap small enough to force an ascent.

The proof is divided into two regimes based on the magnitude of $k$ :

Finite Range ( $4 \le k \le 8,600,001$ ): This section utilizes certified finite computations, explicit estimates for prime-counting functions, and two published prime-record certificates (a large prime gap and a twin prime).
Uniform Tail ( $k \ge 8,600,002$ ): This section employs a uniform construction using the Chinese Remainder Theorem to manufacture a long block of consecutive composite integers, creating a large prime gap. An average-gap argument is then used to identify a smaller subsequent gap within a specific interval.

Key Contributions and Results

Complete Classification: The paper proves that for every integer $k \ge 4$ , the sequence $p \mapsto d_k(p)$ is not unimodal. Combined with Cambie's result for $k \le 3$ , this provides a complete answer to Erdős's question.
- Theorem 1.1: For every $k \ge 4$ , the sequence is not unimodal.
- Corollary 1.2: The sequence is unimodal if and only if $1 \le k \le 3$ .
Explicit Threshold Criterion: The paper establishes a precise algebraic condition (Lemma 3.1) linking the behavior of $d_k(p)$ to the ratio of symmetric polynomials and prime gaps. This reduces the analytic problem to comparing specific numerical values derived from prime distributions.
Certified Computations and Certificates:
- For small $k$ ( $4 \le k \le 20$ ), the authors provide exact certificates using rational arithmetic to compute elementary symmetric sums, identifying specific prime gaps (e.g., $13 \to 17$ for $k=4$ ) that force the required descent/ascent pattern.
- For the range $41 \le k \le 8,600,001$ , the proof leverages two external certificates: a certified large prime gap of length $1,113,106$ and a certified twin prime pair. These are used to demonstrate the existence of the necessary large and small gaps without relying on unproven conjectures.
- A companion numerical verifier is provided to certify all finite decimal inequalities and rational arithmetic used in the proof.
Uniform Construction for Large $k$ : For $k \ge 8,600,002$ , the authors construct a proof that does not rely on specific large prime records. Instead, they use analytic estimates (Dusart's bounds) and the Chinese Remainder Theorem to guarantee the existence of a large gap $G_-$ and a subsequent smaller gap $G_+$ within specific intervals defined by primorials. They rigorously bound the ratio $R_r$ against these gaps to ensure the descent/ascent condition holds.

Significance and Claims
The paper claims to provide the first complete resolution of Erdős Problem 690, settling the question of unimodality for all $k$ . The significance of the work lies in:

Completing the Classification: It definitively answers Erdős's question, showing that unimodality is a property exclusive to the first three cases ( $k=1, 2, 3$ ).
Methodological Demonstration: It presents a proof discovered by the "Multiscalar Fields System" with limited human interaction, demonstrating the capability of automated systems to handle complex number-theoretic proofs involving large-scale computations, analytic estimates, and structural lemmas.
Rigorous Verification: The authors emphasize that the proof is not heuristic; it relies on certified finite computations and explicit analytic bounds, with a publicly available verifier ensuring the correctness of all numerical steps.

The paper does not propose new applications or future experimental directions beyond the resolution of the specific problem and the demonstration of the automated discovery system's efficacy in this domain.