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PreprintCC BY 4.0arXiv

Quantitative lower bound for Pythagorean pairs in boxes of prime powers

  • Rongzhong Xiao
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Abstract

For each positive integer N, let pi be the i-th prime for i∈{1,…,N}, and set ΦN={∏i=1Npiai:ai∈{0,1,…,N−1}}. Write f(N) for the largest cardinality of a subset of ΦN containing no two distinct legs of an integral right triangle. We prove NN−1(logloglog N)≪ f(N).

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Cite

BibTeXRIS
BibTeX
@misc{xiao2026quantitative,
  title = {{Quantitative lower bound for Pythagorean pairs in boxes of prime powers}},
  author = {Rongzhong Xiao},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20311},
  eprint = {2610.20311},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20311},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Quantitative lower bound for Pythagorean pairs in boxes of prime powers
AU  - Rongzhong Xiao
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20311
UR  - https://arxiv.org/abs/2610.20311
AB  - For each positive integer N, let p_i be the i-th prime for i∈{1,…,N}, and set Φ_N={∏_i=1^Np_i^a_i:a_i∈{0,1,…,N−1}}. Write f(N) for the largest cardinality of a subset of Φ_N containing no two distinct legs of an integral right triangle. We prove N^N−1(logloglog N)≪ f(N).
ER  -