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PreprintCC BY-NC-ND 4.0arXiv

On the Diophantine equation Cx2+p2m=yq

  • Usha K. Sangale
  • Priyanka R. Satpute
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Abstract

We classify the primitive positive solutions of Cx2+p2m=yq under the hypotheses that C>3 is odd and square-free, p,q are odd primes, pnmid C, qnmid h(−C), and y is odd. The solutions form six explicit families: one for arbitrary q and five exceptional families with q=3, three of which arise from half-integral elements when C≡3pmod8. The proof combines ideal factorization with Lehmer sequences and shows that representations with second coordinate v=pr, r>0, contribute only when q=3.

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Cite

BibTeXRIS
BibTeX
@misc{sangale2026diophantine,
  title = {{On the Diophantine equation $Cx^2+p^2m=y^q$}},
  author = {Usha K. Sangale and Priyanka R. Satpute},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.02595},
  eprint = {2611.02595},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.02595},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - On the Diophantine equation Cx^2+p^2m=y^q
AU  - Usha K. Sangale
AU  - Priyanka R. Satpute
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.02595
UR  - https://arxiv.org/abs/2611.02595
AB  - We classify the primitive positive solutions of Cx^2+p^2m=y^q under the hypotheses that C>3 is odd and square-free, p,q are odd primes, pnmid C, qnmid h(−C), and y is odd. The solutions form six explicit families: one for arbitrary q and five exceptional families with q=3, three of which arise from half-integral elements when C≡3pmod8. The proof combines ideal factorization with Lehmer sequences and shows that representations with second coordinate v=p^r, r>0, contribute only when q=3.
ER  -