Residue class patterns of consecutive primes

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Cheuk Fung Lau, Residue Class Patterns of Consecutive Primes. The selected text is arXiv:2409.12819v2, revised 13 July 2026. The held PDF is this arXiv version, accompanied by its arXiv record and source sidecars. It was obtained from arXiv. The arXiv record, read 2026-10-15, specifies Creative Commons Attribution 4.0.

Scope

The main theorem concerns a squarefree modulus and a family of residue patterns constructed from an ordered list of reduced residue classes. Its index maps are nondecreasing and have a specified bound on repetitions. It proves that the resulting family is attained infinitely often by consecutive primes. The conclusion does not assert that every prescribed member is attained.

The introduction also gives quantitative lower bounds on the number of attainable patterns. Their parameter restrictions must be retained: for example, the simplified large-range corollary assumes $\varphi(q)>8e^2m\log m$ and sufficiently large $m$.

Read status: claims checked. The abstract and introductory definitions, main theorem and corollaries were read in the arXiv TeX source at statement depth. The sieve proof, the Erdős–Rankin construction and the combinatorial deductions have not been verified here.

Bears on. Problem 251: these statements are a comparison for proposed consecutive-prime residue inputs. The assertions are about attainable families and fixed parameters as $x\to\infty$; they do not supply a uniform estimate for the weighted parity of prime blocks whose length grows with their location.