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PreprintCC BY-NC-ND 4.0arXiv

Ideal class groups of certain real quadratic number fields and factorizations of values of certain quadratic polynomials

  • Stéphane R. Louboutin
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Abstract

We determine the odd primes p for which the number of distinct prime divisors of d−x2 is at most $2$ for every positive odd integers x≤√p. We also determine the odd primes p for which the number of distinct prime divisors of 2p−x2 is at most $2$ for every positive even integers x≤√(2p). To this end, we show that these properties of p and 2p impose restrictions on the class numbers and fundamental units of the real quadratic fields ℚ(√p) and ℚ(√(2p)).

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Cite

BibTeXRIS
BibTeX
@misc{louboutin2026ideal,
  title = {{Ideal class groups of certain real quadratic number fields and factorizations of values of certain quadratic polynomials}},
  author = {Stéphane R. Louboutin},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29347},
  eprint = {2610.29347},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29347},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Ideal class groups of certain real quadratic number fields and factorizations of values of certain quadratic polynomials
AU  - Stéphane R. Louboutin
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29347
UR  - https://arxiv.org/abs/2610.29347
AB  - We determine the odd primes p for which the number of distinct prime divisors of d−x^2 is at most $2$ for every positive odd integers x≤√p. We also determine the odd primes p for which the number of distinct prime divisors of 2p−x^2 is at most $2$ for every positive even integers x≤√(2p). To this end, we show that these properties of p and 2p impose restrictions on the class numbers and fundamental units of the real quadratic fields ℚ(√p) and ℚ(√(2p)).
ER  -