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Equivalence of Restricted and Full Division

  • Naoya Kato
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Abstract

We prove over PA− that the restricted Euclidean division principle pa17−, which asserts division only for co-prime pairs, is equivalent to the full principle pa17. The proof uses a transformation that preserves the quotient in division with remainder to reduce an arbitrary division problem to one for a co-prime pair. This answers Visser's Open Question 8.29 and shows that the restricted principle has the same strength as full Euclidean division over PA−.

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Cite

BibTeXRIS
BibTeX
@misc{kato2026equivalence,
  title = {{Equivalence of Restricted and Full Division}},
  author = {Naoya Kato},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.24976},
  eprint = {2610.24976},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.24976},
}
RIS
TY  - UNPB
TI  - Equivalence of Restricted and Full Division
AU  - Naoya Kato
PY  - 2026
DA  - 2026/10/18
PB  - arXiv
DO  - 10.48550/arxiv.2610.24976
UR  - https://arxiv.org/abs/2610.24976
AB  - We prove over PA^− that the restricted Euclidean division principle pa17^−, which asserts division only for co-prime pairs, is equivalent to the full principle pa17. The proof uses a transformation that preserves the quotient in division with remainder to reduce an arbitrary division problem to one for a co-prime pair. This answers Visser's Open Question 8.29 and shows that the restricted principle has the same strength as full Euclidean division over PA^−.
ER  -