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PreprintCC BY 4.0arXiv

On Diophantine tuples involving star numbers

  • Tapas Chatterjee
  • Sagar Mandal
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Abstract

In this paper, we discuss two related Diophantine tuple problems involving star numbers Sn=6n(n−1)+1. First, we investigate classical D(k)-tuples whose elements are star numbers. We construct infinitely many D(k)-pairs for several positive and negative admissible values of k, and we prove several nonexistence results for D(k)-triples and quadruples. Second, we introduce a modified notion of a star number D(k)-tuple, requiring aiaj+k itself to be a star number. We construct several infinite families of D(1)-triples, and study their extensions to quadruples. In particular, we prove that {4,108,165} has the unique extension {4,108,165,360} to a star number D(1)-quadruple. The main tools used in this paper are generalized Pell equations, Nagell's theorem, lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethő version of the Baker-Davenport reduction method in diophantine approximation.

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Cite

BibTeXRIS
BibTeX
@misc{chatterjee2026diophantine,
  title = {{On Diophantine tuples involving star numbers}},
  author = {Tapas Chatterjee and Sagar Mandal},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27033},
  eprint = {2610.27033},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27033},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Diophantine tuples involving star numbers
AU  - Tapas Chatterjee
AU  - Sagar Mandal
PY  - 2026
DA  - 2026/10/19
PB  - arXiv
DO  - 10.48550/arxiv.2610.27033
UR  - https://arxiv.org/abs/2610.27033
AB  - In this paper, we discuss two related Diophantine tuple problems involving star numbers S_n=6n(n−1)+1. First, we investigate classical D(k)-tuples whose elements are star numbers. We construct infinitely many D(k)-pairs for several positive and negative admissible values of k, and we prove several nonexistence results for D(k)-triples and quadruples. Second, we introduce a modified notion of a star number D(k)-tuple, requiring a_ia_j+k itself to be a star number. We construct several infinite families of D(1)-triples, and study their extensions to quadruples. In particular, we prove that {4,108,165} has the unique extension {4,108,165,360} to a star number D(1)-quadruple. The main tools used in this paper are generalized Pell equations, Nagell's theorem, lower bounds for linear forms in logarithms due to Matveev and Dujella-Pethő version of the Baker-Davenport reduction method in diophantine approximation.
ER  -