RECORD
Residue Class Patterns of Consecutive Primes
Number Theory · Submitted 19 Sep 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this paper, we prove that if $q$ is squarefree, every prescribed sequence of at least $60m\log m$ reduced residue classes mod $q$ contains, in order, an $m$-term block pattern that occurs infinitely often among consecutive primes, with each constant block of length at most $\lceil\log m\rceil$.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Dickson's conjecture and the Hardy--Littlewood prime tuple conjecture predict that every pattern of reduced residue classes modulo $q$ is attained by infinitely many strings of $m$ consecutive primes. At present, however, even proving that a single non-constant residue class pattern of length $m$ occurs infinitely often is beyond the reach of existing methods. Combining Dirichlet's theorem on primes in arithmetic progressions with a theorem of Shiu (2000) shows that, for any $m,q\in\mathbb N$ with $q \ge 3$, at least $m\varphi(q)$ residue class patterns of length $m$ are attained by infinitely many consecutive primes. In this paper, we prove that if $q$ is squarefree, every prescribed sequence of at least $60m\log m$ reduced residue classes mod $q$ contains, in order, an $m$-term block pattern that occurs infinitely often among consecutive primes, with each constant block of length at most $\lceil\log m\rceil$. A recursive combinatorial argument then shows that if $q$ is squarefree and $q \gg (\log m)^2$, then at least \[ \gg \frac{m}{(\log m)^{10}} \varphi(q)^2 \] residue class patterns of length $m$ occur infinitely often among consecutive primes. Moreover, we also show that if $q$ is squarefree and $q \gg (\log m)^2$, then at least \[ \gg e^{-O(m \log_2 m/\log m)} \varphi(q)^{m/\lceil \log m \rceil} \] residue class patterns of length $m$ occur infinitely often among consecutive primes. The proof consists of a modification of the Maynard--Tao sieve found in Banks, Freiberg, and Maynard (2016), by considering the $r$-th moment instead of the 2nd moment for an integer $r$ depending on $m$, which is then combined with an Erd\H{o}s--Rankin type construction.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Dickson's conjecture and the Hardy--Littlewood prime tuple conjecture predict that every pattern of reduced residue classes modulo $q$ is attained by infinitely many strings of $m$ consecutive primes. | background 1.00 | 0.03 | 0.03 | speculative 0.62 |
| s1 | At present, however, even proving that a single non-constant residue class pattern of length $m$ occurs infinitely often is beyond the reach of existing methods. | gap 0.97 | 0.04 | 0.02 | asserted 0.99 |
| s2 | Combining Dirichlet's theorem on primes in arithmetic progressions with a theorem of Shiu (2000) shows that, for any $m,q\in\mathbb N$ with $q \ge 3$, at least $m\varphi(q)$ residue class patterns of length $m$ are attained by infinitely many consecutive primes. | background 0.92 | 0.05 | 0.04 | asserted 1.00 |
| s3 | In this paper, we prove that if $q$ is squarefree, every prescribed sequence of at least $60m\log m$ reduced residue classes mod $q$ contains, in order, an $m$-term block pattern that occurs infinitely often among consecutive primes, with each constant block of length at most $\lceil\log m\rceil$. | result 0.94 | 0.96 | 0.75 | asserted 1.00 |
| s4 | A recursive combinatorial argument then shows that if $q$ is squarefree and $q \gg (\log m)^2$, then at least \[ \gg \frac{m}{(\log m)^{10}} \varphi(q)^2 \] residue class patterns of length $m$ occur infinitely often among consecutive primes. | result 1.00 | 0.94 | 0.17 | asserted 1.00 |
| s5 | Moreover, we also show that if $q$ is squarefree and $q \gg (\log m)^2$, then at least \[ \gg e^{-O(m \log_2 m/\log m)} \varphi(q)^{m/\lceil \log m \rceil} \] residue class patterns of length $m$ occur infinitely often among consecutive primes. | result 0.99 | 0.96 | 0.15 | asserted 1.00 |
| s6 | The proof consists of a modification of the Maynard--Tao sieve found in Banks, Freiberg, and Maynard (2016), by considering the $r$-th moment instead of the 2nd moment for an integer $r$ depending on $m$, which is then combined with an Erd\H{o}s--Rankin type construction. | method 1.00 | 0.82 | 0.03 | asserted 0.99 |
What was examined, and what was not
- arXiv listing abstract v2
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Dickson's conjecture and the Hardy--Littlewood prime tuple conjecture predict that every pattern of reduced residue classes modulo $q$ is attained by infinitely many strings of $m$ consecutive primes. At present, however, even proving that a single non-constant residue class pattern of length $m$ occurs infinitely often is beyond the reach of existing methods. Combining Dirichlet's theorem on primes in arithmetic progressions with a theorem of Shiu (2000) shows that, for any $m,q\in\mathbb N$ with $q \ge 3$, at least $m\varphi(q)$ residue class patterns of length $m$ are attained by infinitely many consecutive primes. In this paper, we prove that if $q$ is squarefree, every prescribed sequence of at least $60m\log m$ reduced residue classes mod $q$ contains, in order, an $m$-term block pattern that occurs infinitely often among consecutive primes, with each constant block of length at most $\lceil\log m\rceil$. A recursive combinatorial argument then shows that if $q$ is squarefree and $q \gg (\log m)^2$, then at least \[ \gg \frac{m}{(\log m)^{10}} \varphi(q)^2 \] residue class patterns of length $m$ occur infinitely often among consecutive primes. Moreover, we also show that if $q$ is squarefree and $q \gg (\log m)^2$, then at least \[ \gg e^{-O(m \log_2 m/\log m)} \varphi(q)^{m/\lceil \log m \rceil} \] residue class patterns of length $m$ occur infinitely often among consecutive primes. The proof consists of a modification of the Maynard--Tao sieve found in Banks, Freiberg, and Maynard (2016), by considering the $r$-th moment instead of the 2nd moment for an integer $r$ depending on $m$, which is then combined with an Erd\H{o}s--Rankin type construction.
Human review
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.