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On continued fraction partial quotients of square roots of primes

continued fraction

prime number

square root of a positive integer

We show that for each positive integer a there exist only finitely many prime numbers p such that a appears an odd number of times in the period of continued fraction of or . We also prove that if p is a prime number and or 2p is such that the length of the period of continued fraction expansion of is divisible by 4, then 1 appears as a partial quotient in the continued fraction of . Furthermore, we give an upper bound for the period length of continued fraction expansion of , where D is a positive non-square, and factorize some family of polynomials with integral coefficients connected with continued fractions of square roots of positive integers. These results answer several questions recently posed by Miska and Ulas [MU].

dc.abstract.en | We show that for each positive integer a there exist only finitely many prime numbers p such that a appears an odd number of times in the period of continued fraction of or . We also prove that if p is a prime number and or 2p is such that the length of the period of continued fraction expansion of is divisible by 4, then 1 appears as a partial quotient in the continued fraction of . Furthermore, we give an upper bound for the period length of continued fraction expansion of , where D is a positive non-square, and factorize some family of polynomials with integral coefficients connected with continued fractions of square roots of positive integers. These results answer several questions recently posed by Miska and Ulas [MU]. | pl |

dc.affiliation | Wydział Matematyki i Informatyki : Instytut Matematyki | pl |

dc.contributor.author | Kala, Vítězslav | pl |

dc.contributor.author | Miska, Piotr - 191205 | pl |

dc.date.accessioned | 2023-08-21T11:15:55Z | |

dc.date.available | 2023-08-21T11:15:55Z | |

dc.date.issued | 2023 | pl |

dc.date.openaccess | 0 | |

dc.description.accesstime | w momencie opublikowania | |

dc.description.physical | 215-234 | pl |

dc.description.version | ostateczna wersja wydawcy | |

dc.description.volume | 253 | pl |

dc.identifier.doi | 10.1016/j.jnt.2023.06.013 | pl |

dc.identifier.eissn | 1096-1658 | pl |

dc.identifier.issn | 0022-314X | pl |

dc.identifier.uri | https://ruj.uj.edu.pl/xmlui/handle/item/317828 | |

dc.language | eng | pl |

dc.language.container | eng | pl |

dc.rights | Udzielam licencji. Uznanie autorstwa 4.0 Międzynarodowa | * |

dc.rights.licence | CC-BY | |

dc.rights.uri | http://creativecommons.org/licenses/by/4.0/legalcode.pl | * |

dc.share.type | inne | |

dc.subject.en | continued fraction | pl |

dc.subject.en | prime number | pl |

dc.subject.en | square root of a positive integer | pl |

dc.subtype | Article | pl |

dc.title | On continued fraction partial quotients of square roots of primes | pl |

dc.title.journal | Journal of Number Theory | pl |

dc.type | JournalArticle | pl |

dspace.entity.type | Publication |