Frobenius Numbers Associated with Primitive Pythagorean Quadruples

Abstract

Let (a,b,c,d) = (2mn, 2mp, m² − n² − p², m² + n² + p²) be a primitive Pythagorean quadruple and let S=⟨ a, b, c, d⟩ be the numerical semigroup generated by a,b,c, and d. For convenience, we also let Q = n² + p², δ= gcd(n,p), and n = δn₀ for some n₀ ∈ ℤ. In this paper, we determine the Frobenius number of S and derive an explicit formula in terms of m, n, and p with the assumption that m≥ 2Q for δ= 1 and m ≥ 2Q/δ − 1 for the remaining cases. The proof is based on an explicit complete residue system modulo 2mn, a normalization procedure for arbitrary semigroup elements, and a lift-orbit description of boundary representatives.

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BibTeX RIS

BibTeX

@misc{hwang2026frobenius,
  title = {{Frobenius Numbers Associated with Primitive Pythagorean Quadruples}},
  author = {WonTae Hwang and Kyunghwan Song},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.21397},
  eprint = {2609.21397},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.21397},
}

RIS

TY  - UNPB
TI  - Frobenius Numbers Associated with Primitive Pythagorean Quadruples
AU  - WonTae Hwang
AU  - Kyunghwan Song
PY  - 2026
DA  - 2026/09/18
PB  - arXiv
DO  - 10.48550/arxiv.2609.21397
UR  - https://arxiv.org/abs/2609.21397
AB  - Let (a,b,c,d) = (2mn, 2mp, m^2 − n^2 − p^2, m^2 + n^2 + p^2) be a primitive Pythagorean quadruple and let S=⟨ a, b, c, d⟩ be the numerical semigroup generated by a,b,c, and d. For convenience, we also let Q = n^2 + p^2, δ= gcd(n,p), and n = δn_0 for some n_0 ∈ ℤ. In this paper, we determine the Frobenius number of S and derive an explicit formula in terms of m, n, and p with the assumption that m≥ 2Q for δ= 1 and m ≥ 2Q/δ − 1 for the remaining cases. The proof is based on an explicit complete residue system modulo 2mn, a normalization procedure for arbitrary semigroup elements, and a lift-orbit description of boundary representatives.
ER  -

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