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PreprintCC BY 4.0arXiv

Goldbach and Alexandroff Shake Hands

  • Lourival Lima
  • Jorge Vielma
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Abstract

We introduce a relational Alexandroff topology associated with the strong Goldbach conjecture on the set of even integers greater than or equal to six. The construction is induced by a multivalued map encoding Goldbach representations and leads naturally to a notion of relational reachability. For arbitrary binary relations, we show that singleton closures coincide with reachable sets and that the common intersection of all singleton closures consists exactly of the globally reachable points. In the Goldbach setting, this yields a topological characterization of the conjecture in terms of the distinguished point 6. In particular, the strong Goldbach conjecture is equivalent to the associated relational space being non-wR0 and connected. We also obtain compactness and supercompactness criteria in terms of the set of possible Goldbach exceptions.

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Cite

BibTeXRIS
BibTeX
@misc{lima2026goldbach,
  title = {{Goldbach and Alexandroff Shake Hands}},
  author = {Lourival Lima and Jorge Vielma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.25573},
  eprint = {2610.25573},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.25573},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Goldbach and Alexandroff Shake Hands
AU  - Lourival Lima
AU  - Jorge Vielma
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.25573
UR  - https://arxiv.org/abs/2610.25573
AB  - We introduce a relational Alexandroff topology associated with the strong Goldbach conjecture on the set of even integers greater than or equal to six. The construction is induced by a multivalued map encoding Goldbach representations and leads naturally to a notion of relational reachability. For arbitrary binary relations, we show that singleton closures coincide with reachable sets and that the common intersection of all singleton closures consists exactly of the globally reachable points. In the Goldbach setting, this yields a topological characterization of the conjecture in terms of the distinguished point 6. In particular, the strong Goldbach conjecture is equivalent to the associated relational space being non-wR_0 and connected. We also obtain compactness and supercompactness criteria in terms of the set of possible Goldbach exceptions.
ER  -