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On the Number of Prime Factors of Consecutive Integers

This paper proves that there are infinitely many integers $n$ such that the number of distinct prime factors of all subsequent integers $n+k$ is bounded by $O(\log k)$ , significantly improving upon a previous $O(k)$ bound by Tao and Teräväinen and making progress on several questions posed by Erdős.

Original authors: Cheuk Fung Lau

Published 2026-04-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Cheuk Fung Lau

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 are walking down a long, endless street of houses, numbered 1, 2, 3, and so on. Each house represents an integer. Now, imagine that every house has a "security team" inside it. The size of this team is determined by the number of distinct prime factors the house number has.

A house numbered 6 (which is $2 \times 3$ ) has a small team of 2.
A house numbered 30 (which is $2 \times 3 \times 5$ ) has a team of 3.
A house numbered 2,310 (which is $2 \times 3 \times 5 \times 7 \times 11$ ) has a team of 5.

Usually, as the house numbers get bigger, these security teams get larger and larger. It's rare to find a long stretch of houses where the teams stay small.

The Big Question

The mathematician Paul Erdős (a legendary puzzle master) asked a very specific question: Is it possible to find a starting house number $n$ such that for a long stretch of houses following it ( $n+1, n+2, n+3...$ ), the security teams remain surprisingly small?

Specifically, he wondered if the team size could stay below a certain limit that grows very slowly as you move down the street.

The Previous Answer (The "Good" News)

In 2025, mathematicians Terence Tao and Jori Teräväinen proved that yes, such stretches do exist infinitely often. However, their proof allowed the team size to grow quite a bit. They showed the team size could be roughly proportional to the distance you walked ( $k$ ).

Think of it like this: If you walk 100 houses down the street, their proof said the security team could be as big as 100 people. That's a lot of people!

The New Breakthrough (The "Better" News)

In this new paper, Joshua Lau (the author) has improved that result significantly. He proved that we can find stretches where the security team size is much, much smaller.

Instead of the team size growing like the distance ( $k$ ), Lau proved it only needs to grow like the logarithm of the distance ( $\log k$ ).

The Analogy:

Tao & Teräväinen (2025): If you walk 1,000,000 houses, the security team might be 1,000,000 people.
Joshua Lau (2026): If you walk 1,000,000 houses, the security team is only about 14 people (since $\log_{10} 1,000,000 = 6$ , and the math works out to a small multiple of that).

This is a massive improvement. It means we can find incredibly long sequences of numbers where the "complexity" (number of prime factors) stays remarkably low.

How Did He Do It? (The Magic Trick)

Lau didn't just look at numbers one by one. He used a probabilistic approach, which is like a high-tech sieve.

The Sieve: Imagine a giant sieve (a colander) that filters out numbers with "bad" properties (like having too many small prime factors).
The Weights: Instead of just keeping or throwing away numbers, Lau assigned them "weights" (scores). He gave higher scores to numbers that looked like they had few prime factors.
The Random Walk: He treated the selection of the starting number $n$ as a random event, but one that was heavily biased by his sieve.
The Concentration: The hardest part was proving that all the numbers in the sequence ( $n+1, n+2, \dots$ ) would behave well at the same time. Usually, if you get lucky with one, you might get unlucky with the next. Lau used advanced math (called "concentration of measure") to show that if you set up the sieve correctly, the whole group behaves like a well-organized choir rather than a chaotic crowd.

The "Best Possible" Guess

Lau also made a bold guess (a conjecture) based on how random numbers behave. He believes his new result is essentially the best possible.

He argues that you can't push the team size down much further than $\log k$ . If you try to demand the teams be even smaller (like $\log k$ divided by 2), the math suggests such sequences simply don't exist. It's like saying, "You can't fit 100 people in a car that only has 5 seats, no matter how you arrange them."

Why Does This Matter?

This isn't just about counting factors. It helps us understand the hidden structure of numbers.

Erdős's Problems: This paper solves or makes huge progress on several famous problems posed by Erdős (specifically problems #248, #413, #826, and #679 on his problem list).
The "Falsity" Claim: The paper also suggests that one of Erdős's older, very optimistic guesses (that the team size could be even smaller than Lau's result) is likely false. Lau uses a "random model" to show that if you assume primes behave like random coins flipping, Erdős's guess breaks down.

Summary

Joshua Lau has shown that there are infinitely many "quiet neighborhoods" in the world of numbers where the complexity of the houses stays very low for a long time. He improved the previous record by a huge margin, using a sophisticated mix of probability and sieving, and he believes he has found the absolute limit of how quiet these neighborhoods can get.

1. Problem Statement and Context

The paper addresses a fundamental question in analytic number theory regarding the distribution of prime factors in sequences of consecutive integers. Specifically, it investigates the existence of integers $n$ such that the numbers $n+k$ (for $k \ge 1$ ) simultaneously possess a "small" number of distinct prime factors, denoted by $\omega(n+k)$ .

The author refines a recent result by Tao and Teräväinen (2025) concerning Erdős's Conjecture 1 (Erdős Problem #248).

Previous Result (Tao-Teräväinen): There exist infinitely many $n$ such that $\omega(n+k) \ll k$ for all $k \ge 1$ .
The Gap: The bound $O(k)$ is linear in $k$ . Erdős conjectured that the bound could be significantly tighter, specifically logarithmic in $k$ .
Goal: To prove that there are infinitely many $n$ such that $\omega(n+k) \ll \log k$ for all $k \ge 2$ , which is a near-optimal bound given probabilistic models.

2. Methodology

The proof employs a probabilistic sieve method, building upon the framework established by Tao and Teräväinen but introducing significant quantitative refinements.

A. Probabilistic Construction

The author constructs a random variable $n$ taking values in the interval $[x, 2x]$ with a specific probability distribution. Instead of the uniform distribution, $n$ is weighted by a product of Goldston–Pintz–Yıldırım (GPY) type sieve weights:
$w(n) \propto \mathbb{1}_{|n \forall p \le 0.15 \log x} \prod_{k=1}^{K} \left( \sum_{\substack{d | n+k \\ (d, P(w))=1}} \mu(d) \tilde{\eta}\left(\frac{\log d}{\log R_k}\right) \right)^2$
Here:

$K = (\log x)^{1/1000}$ .
$R_k$ are sieve levels chosen to decay polynomially: $R_k = x^{1/(100k^{50})}$ for small $k$ .
$\tilde{\eta}$ is a smooth function from the Gevrey class, used to localize the sieve.

B. Quantitative Refinements

The core innovation lies in the choice of sieve levels $R_k$ and the analysis of moments:

Polynomial Decay of Sieve Levels: Unlike the constant or slower decay in previous works, the author chooses $R_k$ to decay as $x^{1/k^{50}}$ . This balances the requirement that the product of sieve levels $\prod R_k < x^\theta$ (to avoid overlap issues) while keeping individual $R_k$ large enough to suppress small prime factors effectively.
Concentration of Measure: The proof relies on establishing strong exponential concentration-of-measure estimates. The author proves that under the weighted distribution, the random variable $\omega(n+k)$ satisfies a Central Limit Theorem-type behavior with mean and variance proportional to $\log(\log x / \log R_k) \approx \log k$ .
Higher Moment Bounds: To achieve the $\log k$ $lo g k$ bound, the author derives upper bounds for high-order moments ( $s$ $s$ -th moments) of the number of prime factors. This allows the use of Chebyshev-type inequalities with much tighter tail bounds than the second-moment estimates used by Tao and Teräväinen.
- Specifically, the probability of the tail event $\omega(n+k) \ge C \log k$ is shown to be $\ll k^{-cC}$ . By choosing $C$ large enough, the sum of these probabilities over all $k$ converges to less than 1.

C. Technical Tools

Fourier Analysis: The sieve weights are analyzed using Fourier inversion and properties of the Gevrey class functions to control error terms.
Stirling Numbers: Used to expand moments of sums of indicator functions.
Euler Products: Local factors in the sieve are analyzed via Euler products to estimate the probability of divisibility by specific primes.

3. Key Results

Main Theorem (Theorem 1.1)

There exists a positive constant $C$ such that for infinitely many positive integers $n$ :
$\omega(n+k) \le \Omega(n+k) \le C \log k \quad \text{for every integer } k \ge 2.$
This improves the previous $O(k)$ bound to $O(\log k)$ .

Corollaries and Extensions

Divisor Function (Corollary 1.2): As a consequence, there are infinitely many $n$ such that $\tau(n+k) \ll k^C$ for all $k \ge 1$ (a weaker version of Erdős's Conjecture 3).
Backward Sequence (Theorem 1.3): The result holds symmetrically for $n-k$ : there are infinitely many $n$ such that $\omega(n-k) \le C \log k$ for $1 < k < n$ .
Optimality Conjecture (Conjecture 5 & 6): Based on Cramér-type random models, the author conjectures that the $C \log k$ bound is essentially sharp. Specifically, for any $\epsilon > 0$ , there exist $k$ such that $\omega(n+k) > (1-\epsilon) \log k$ .

4. Conditional Falsity of Erdős Problem #679

The paper addresses Erdős Problem #679 (Conjecture 4), which posits that for any $\epsilon > 0$ , there are infinitely many $n$ such that $\omega(n-k) \le (1+\epsilon) \frac{\log k}{\log \log k}$ .

The Author's Conjecture (Conjecture 7/8): Based on Cramér's model for primes, the author conjectures that integers with "many" prime factors (specifically $\omega(n) \ge C_0 \frac{\log_2 n}{\log_3 n}$ ) appear frequently enough that they cannot be avoided in short intervals.
Implication: Assuming this conjecture, the author proves Theorem 7.3: There exists $\delta > 0$ such that for every sufficiently large $n$ , there exists $k$ where $\omega(n-k) > (1+\delta) \frac{\log k}{\log \log k}$ .
Conclusion: This implies that the bound in Erdős's Conjecture 4 is false; the "true" lower bound for the maximum number of prime factors in a short interval is likely higher than $(1+\epsilon) \frac{\log k}{\log \log k}$ .

5. Significance

Resolution of a Major Open Problem: The paper provides a significant improvement on a problem posed by Erdős in 1974, moving the bound from linear to logarithmic.
Methodological Advancement: The combination of GPY sieves with refined polynomial decay of sieve levels and high-moment concentration estimates represents a powerful new technique in probabilistic number theory.
Insight into Prime Distribution: The work suggests that the distribution of prime factors in consecutive integers is governed by probabilistic laws that are tighter than previously proven, and it challenges existing conjectures about the lower bounds of prime factors in short intervals.
Sharpness: The formulation of Conjecture 5 provides a clear target for future research, suggesting that the author's bound is likely the best possible up to a constant factor.

In summary, Lau's paper represents a major leap forward in understanding the local behavior of the prime factor counting function $\omega(n)$ , utilizing sophisticated probabilistic sieving to prove that consecutive integers can simultaneously have very few prime factors, specifically bounded logarithmically in their distance from $n$ .