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On the Diophantine equation X2+pm=p1n1p2n2

  • Amir Ghadermarzi
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Abstract

We consider the Diophantine equation X2+pm=p1n1p2n2, where p,p1,p2 are distinct odd primes. Using factorizations in quadratic fields, sharp bounds for linear forms in two p-adic logarithms, elementary methods, and known results on exponential Diophantine equations, we obtain explicit bounds for m,n1,n2, and consequently the finiteness of the number of solutions for fixed p,p1,p2.

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Cite

BibTeXRIS
BibTeX
@misc{ghadermarzi2026diophantine,
  title = {{On the Diophantine equation $X^2+p^m=p_1^n_1p_2^n_2$}},
  author = {Amir Ghadermarzi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20930},
  eprint = {2610.20930},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20930},
}
RIS
TY  - UNPB
TI  - On the Diophantine equation X^2+p^m=p_1^n_1p_2^n_2
AU  - Amir Ghadermarzi
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20930
UR  - https://arxiv.org/abs/2610.20930
AB  - We consider the Diophantine equation X^2+p^m=p_1^n_1p_2^n_2, where p,p_1,p_2 are distinct odd primes. Using factorizations in quadratic fields, sharp bounds for linear forms in two p-adic logarithms, elementary methods, and known results on exponential Diophantine equations, we obtain explicit bounds for m,n_1,n_2, and consequently the finiteness of the number of solutions for fixed p,p_1,p_2.
ER  -