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Residue Class Patterns of Consecutive Primes

This paper establishes that for squarefree moduli $q$ , a significant number of residue class patterns of length $m$ occur infinitely often among consecutive primes by combining a modified Maynard–Tao sieve with an Erdős–Rankin construction to prove the existence of specific block patterns and derive improved lower bounds on the count of such patterns.

Original authors: Cheuk Fung Lau

Published 2026-07-14

📖 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 the prime numbers as a row of VIP guests arriving at an exclusive, infinite party. Each guest wears a badge with a number on it, but the number is only visible if you look at it through a specific colored filter (a modulus, let's call it $q$ ). When you look through this filter, the badge numbers wrap around, like a clock.

For a long time, mathematicians have suspected that these VIPs (primes) are incredibly random. They think that if you look at a long line of consecutive guests, you should see every possible combination of badge colors eventually, and you should see them over and over again, forever. This is the "Hardy–Littlewood prime tuple conjecture." It's a beautiful idea, but it's currently just a hunch. We can't prove it yet. In fact, even proving that one specific, non-repeating pattern of badge colors happens infinitely often is currently beyond our reach.

However, in this paper, Cheuk Fung (Joshua) Lau has managed to find a way to prove that we can find a huge number of these patterns happening infinitely often, provided we have enough different badge colors to choose from.

The "Block Party" Discovery

Think of the sequence of primes as a long line of people. Lau's main result is like finding a massive, organized block party within this infinite line.

He proves that if you have a "squarefree" modulus $q$ (think of this as a filter that doesn't have any repeated prime factors, like a clean, simple lens) and the number of available badge colors ( $\phi(q)$ ) is large enough, then you can find a specific sequence of $m$ consecutive primes that follows a prescribed pattern of colors.

Here is the magic trick: Lau doesn't just find one pattern. He shows that if you write down a very long list of at least $60m \log m$ prescribed residue classes (badge colors), this long list is guaranteed to contain, in order, a block of $m$ consecutive primes that matches a pattern that repeats infinitely often.

To make this work, he had to be a bit flexible. He showed that within this long list, you can find a pattern where the colors stay the same for short bursts (blocks of length at most $\lceil \log m \rceil$ ) before switching to a new color. It's like finding a dance routine where the dancers hold a pose for a few seconds, then switch to a new pose, and this specific routine happens over and over again in the infinite line of primes.

The Tools: A Sieve and a Construction

How did he do it? He combined two powerful mathematical tools.

First, he used a modified version of the Maynard–Tao sieve. Imagine this as a super-fine net designed to catch clusters of primes that are very close together. Usually, this net catches primes that are just "somewhat" close. Lau tweaked the net (by looking at the $r$ -th moment instead of just the 2nd) to ensure that the primes he caught were not just close, but actually consecutive in the line of all primes.

Second, he used a modified Erdős–Rankin construction. This is like a master architect's blueprint. He used it to build a specific "trap" (a set of numbers) that forces the primes caught by the sieve to wear the exact badge colors he wanted. He carefully arranged the trap so that the primes would land in the right "residue classes" (colors) modulo $q$ .

The Results: How Many Patterns?

The paper explicitly rules out the idea that we can currently prove every possible pattern happens infinitely often. That remains a conjecture. Lau's work is a massive step forward, but it doesn't solve the whole mystery.

Instead, he gives us a lower bound—a guarantee of how many patterns we know happen infinitely often.

The "Medium" Range: If the number of available colors $\phi(q)$ is larger than roughly $3823e^{10}(\log m)^{10} + 1$ , then Lau proves there are at least a specific, huge number of patterns that occur infinitely often. This number is roughly proportional to $\phi(q)(\phi(q)-1)$ . This is a significant improvement over previous methods, which could only guarantee about $m\phi(q)$ patterns.
The "Large" Range: If the number of colors is even larger (specifically, if $\phi(q) > 8e^2 m \log m$ ), the number of guaranteed patterns explodes. The paper shows there are at least roughly $e^{-O(m \log^2 m / \log m)} \phi(q)^{m / \lceil \log m \rceil}$ patterns.

The "Shifting" Trick

To get these numbers, Lau uses a clever combinatorial argument. He starts with a "good" pattern found by his sieve and then "shifts" it. Imagine you have a valid dance routine. If you take the last dancer and move them to the front, or shift the whole line, you might get a new, valid routine. By repeating this shifting process, he multiplies the number of guaranteed patterns.

However, he is careful to note that this shifting only works if the patterns are distinct. He proves that by shifting a pattern at most $\lfloor (1-c)m \rfloor$ times, you get a unique new pattern every time, provided the original pattern was constructed carefully.

The Bottom Line

This paper doesn't prove that every pattern of prime badge colors happens infinitely often. That dream is still out of reach. But it does prove that if you have a sufficiently large set of colors (specifically, if $q$ is squarefree and $\phi(q)$ is large enough), then a vast number of specific, non-constant patterns are guaranteed to appear infinitely often in the line of consecutive primes.

The authors are very sure about this because they have a rigorous proof, not just a simulation or a guess. They have constructed the mathematical machinery (the modified sieve and the construction) to demonstrate that these patterns must exist. While we can't yet see the full rainbow of prime patterns, Lau has successfully proven that a massive, colorful chunk of that rainbow is definitely there, repeating forever.

Technical Summary: Residue Class Patterns of Consecutive Primes

Problem Statement
The paper addresses a fundamental question in analytic number theory regarding the distribution of consecutive primes modulo an integer $q$ . While the Hardy–Littlewood prime tuple conjecture and Dickson's conjecture predict that every admissible pattern of reduced residue classes modulo $q$ occurs infinitely often among $m$ consecutive primes, current methods are insufficient to prove even the existence of a single non-constant pattern of length $m > 1$ occurring infinitely often.

Existing results are limited:

Dirichlet's Theorem: Guarantees infinitely many primes in any single residue class ( $m=1$ ).
Shiu (2000): Proved that constant patterns $(a, a, \dots, a)$ occur infinitely often for any $a \in (\mathbb{Z}/q\mathbb{Z})^\times$ .
Current Gap: It is unknown if any specific non-constant pattern (e.g., $(a, b)$ with $a \neq b$ ) occurs infinitely often.

The paper aims to establish a lower bound on the number of residue class patterns of length $m$ that are attained by infinitely many consecutive primes, specifically for squarefree moduli $q$ .

Methodology
The proof combines two primary components: a modification of the Maynard–Tao sieve and a modified Erdős–Rankin type construction.

Modified Maynard–Tao Sieve (Section 5):
- The authors adapt the sieve weights introduced by Banks, Freiberg, and Maynard (2016).
- Instead of focusing solely on the second moment (counting pairs of primes), the method considers the $r$ -th moment for an integer $r > 1$ depending on $m$ .
- The sieve is designed to detect clusters of primes within a specific admissible $k$ -tuple $H = \{h_1, \dots, h_k\}$ . The goal is to find an integer $n$ such that the shifted tuple $qn + H$ contains primes in specific "blocks" while leaving gaps between them.
- Specifically, the sieve guarantees the existence of $n$ such that a partition of $H$ into $M$ subsets yields $m+1$ subsets containing between $1$ and $r-1$ primes, with no primes in the intermediate subsets. This structure ensures the detected primes are consecutive.
Modified Erdős–Rankin Construction (Section 6):
- To ensure the primes detected by the sieve lie in prescribed residue classes modulo $q$ , the authors construct an admissible $k$ -tuple $\{h_1, \dots, h_k\}$ with specific properties.
- Using a variation of the Erdős–Rankin construction, they select $h_i$ such that $h_i \equiv r_i \pmod q$ for prescribed residues $r_i$ .
- Crucially, the construction ensures that the pairwise differences $h_i - h_j$ are "smooth" (having only small prime factors), a necessary condition for the Maynard–Tao sieve to function effectively.
- This construction also ensures that the primes found in the interval $(qn, qn+z]$ are consecutive, as all integers in the gap are sieved out by small primes.
Combinatorial Argument (Section 8):
- Once the existence of specific block patterns is established via the sieve and construction, the authors use a recursive combinatorial argument to count the total number of distinct $m$ -tuples that occur infinitely often.
- They define "good" tuples based on the constraints of the sieve (piecewise constant blocks with no $r$ consecutive identical values) and use a shifting argument to generate distinct patterns from a base set.

Key Contributions and Results

Main Theorem (Theorem 1.5): For a squarefree integer $q$ , and any sequence of $M$ prescribed residue classes $a_1, \dots, a_M \in (\mathbb{Z}/q\mathbb{Z})^\times$ , there exist infinitely many consecutive primes whose residue class pattern matches a sequence formed by these $a_i$ 's, subject to the constraint that no $r$ consecutive terms in the pattern are identical. Here, $M$ is a parameter depending on $m$ and $r$ .
Lower Bounds for Medium $q$ (Corollary 1.6): If $q$ is squarefree and $\phi(q) \gg (\log m)^2$ , the paper proves that at least
$\gg \frac{m}{(\log m)^{10}} \phi(q)^2$
residue class patterns of length $m$ occur infinitely often. This improves upon the trivial lower bound of $m\phi(q)$ derived from Shiu's theorem when $\phi(q)$ is sufficiently large relative to $m$ .
Lower Bounds for Large $q$ (Corollary 1.8): If $q$ is squarefree and $\phi(q) \gg m \log m$ , the number of attainable patterns is at least
$\gg e^{-O(m \log^2 m / \log m)} \phi(q)^{m / \lceil \log m \rceil}.$
This result approaches the conjectured total number of patterns $\phi(q)^m$ more closely as $m$ increases, though it still falls short of the full conjecture.

Significance and Claims
The paper claims to make the first significant progress beyond the constant-pattern results of Shiu (2000) and Maynard (2016). By proving that a vast number of non-constant patterns occur infinitely often, the work provides strong evidence supporting the heuristic that consecutive primes behave randomly modulo $q$ , aside from local congruence obstructions.

The authors explicitly state that proving a single specific non-constant pattern occurs infinitely often remains "beyond the reach of existing methods." Therefore, the significance of this work lies in establishing a quantitative lower bound on the set of attainable patterns, demonstrating that the set of known attainable patterns is much larger than previously established, rather than identifying a specific new pattern. The results are conditional on $q$ being squarefree and sufficiently large relative to $m$ .

The "Block Party" Discovery

The Tools: A Sieve and a Construction

The Results: How Many Patterns?

The "Shifting" Trick

The Bottom Line

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