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PreprintCC BY 4.0Zenodo

Long Composite Stretches at Consecutive Prime-Gap Products: A Uniform Maynard Argument

  • Riccardo Sparacino
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Abstract

For consecutive primes p0 and D. A single fixed gap occurs for infinitely many such pairs. The proof controls the growing CRT modulus: a presieve leaves O(L/log L) targets, giving log Q=O(L) for L proportional to log log X. With m=2 and ε=1/100 fixed, the resulting coefficient Q≤(log T)^(1/4) lies within the coefficient range of Maynard’s Theorem 3.4. This result is compatible with K(M)≤ceil((log M)^2); it neither proves nor disproves the universal Sparacino conjecture. This note expands Theorem 2 in the author’s earlier version 13 (https://doi.org/10.5281/zenodo.23638603) and makes no priority claim.

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Cite

BibTeXRIS
BibTeX
@misc{sparacino2026long,
  title = {{Long Composite Stretches at Consecutive Prime-Gap Products: A Uniform Maynard Argument}},
  author = {Riccardo Sparacino},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23641757},
  url = {https://zenodo.org/records/23641757},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Long Composite Stretches at Consecutive Prime-Gap Products: A Uniform Maynard Argument
AU  - Riccardo Sparacino
PY  - 2026
DA  - 2026/10/25
PB  - Zenodo
DO  - 10.5281/zenodo.23641757
UR  - https://zenodo.org/records/23641757
AB  - For consecutive primes p0 and D. A single fixed gap occurs for infinitely many such pairs. The proof controls the growing CRT modulus: a presieve leaves O(L/log L) targets, giving log Q=O(L) for L proportional to log log X. With m=2 and ε=1/100 fixed, the resulting coefficient Q≤(log T)^(1/4) lies within the coefficient range of Maynard’s Theorem 3.4. This result is compatible with K(M)≤ceil((log M)^2); it neither proves nor disproves the universal Sparacino conjecture. This note expands Theorem 2 in the author’s earlier version 13 (https://doi.org/10.5281/zenodo.23638603) and makes no priority claim.
ER  -