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PreprintCC BY 4.0arXiv

Prime Quadruplets and Jump Conditions on Arithmetic Functions

  • Himaghna Roy Choudhury
  • Shicheng Wei
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Abstract

We provide progress on the characterization of composite integers n that satisfy the jump conditions φ(n+12)=φ(n)+12 and σ(n+12)=σ(n)+12 simultaneously. While it is known that prime quadruplets (p,p+2,p+6,p+8) generate solutions n=p(p+8), the complete characterization remains an open conjecture. We prove that this characterization is complete when n and n+12 are both squarefree semiprimes, and that no solution n can be a prime power. Furthermore, a complete search up to 1012 resulted in no counterexamples to the conjecture. If this conjecture is proven true, and there are infinitely many such solutions, then it can be proved that there are infinitely many prime quadruplets.

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Cite

BibTeXRIS
BibTeX
@misc{choudhury2026prime,
  title = {{Prime Quadruplets and Jump Conditions on Arithmetic Functions}},
  author = {Himaghna Roy Choudhury and Shicheng Wei},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.10331},
  eprint = {2606.10331},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.10331},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Prime Quadruplets and Jump Conditions on Arithmetic Functions
AU  - Himaghna Roy Choudhury
AU  - Shicheng Wei
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2606.10331
UR  - https://arxiv.org/abs/2606.10331
AB  - We provide progress on the characterization of composite integers n that satisfy the jump conditions φ(n+12)=φ(n)+12 and σ(n+12)=σ(n)+12 simultaneously. While it is known that prime quadruplets (p,p+2,p+6,p+8) generate solutions n=p(p+8), the complete characterization remains an open conjecture. We prove that this characterization is complete when n and n+12 are both squarefree semiprimes, and that no solution n can be a prime power. Furthermore, a complete search up to 10^12 resulted in no counterexamples to the conjecture. If this conjecture is proven true, and there are infinitely many such solutions, then it can be proved that there are infinitely many prime quadruplets.
ER  -