대한수학회보 (Bulletin of the Korean Mathematical Society)

  • 제59권3호
  • /
  • Pages.697-707
  • /
  • 2022
  • /
  • 1015-8634(pISSN)
  • /
  • 2234-3016(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

EXISTENCE OF THE CONTINUED FRACTIONS OF ${\sqrt{d}}$ AND ITS APPLICATIONS

  • Lee, Jun Ho (Department of Mathematics Education Mokpo National University)
  • 투고 : 2021.05.31
  • 심사 : 2022.02.04
  • 발행 : 2022.05.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

It is well known that the continued fraction expansion of ${\sqrt{d}}$ has the form $[{\alpha}_0,\;{\bar{{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},\;2_{{\alpha}_0}}]$ and ${\alpha}_1,\;{\ldots},\;{\alpha}_{l-1}$ is a palindromic sequence of positive integers. For a given positive integer l and a palindromic sequence of positive integers ${\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},$ we define the set $S(l;{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1})\;:=\;\{d{\in}{\mathbb{Z}}{\mid}d>0,\;{\sqrt{d}}=[{\alpha}_0,\;{\bar{{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1},\;2_{{\alpha}_0}}],\;where\;{\alpha}_0={\lfloor}{\sqrt{d}}{\rfloor}\}.$ In this paper, we completely determine when $S(l;{\alpha}_1,\;{\ldots},\;{\alpha}_{l-1})$ is not empty in the case that l is 4, 5, 6, or 7. We also give similar results for $(1+{\sqrt{d}})/2.$ For the case that l is 4, 5, or 6, we explicitly describe the fundamental units of the real quadratic field ${\mathbb{Q}}({\sqrt{d}}).$ Finally, we apply our results to the Mordell conjecture for the fundamental units of ${\mathbb{Q}}({\sqrt{d}}).$

키워드

과제정보

The author sincerely thanks the referees for their valuable comments which improved the original version of this manuscript.

참고문헌

  1. N. C. Ankeny, E. Artin, and S. Chowla, The class-number of real quadratic number fields, Ann. of Math. (2) 56 (1952), 479-493. https://doi.org/10.2307/1969656
  2. T. Azuhata, On the fundamental units and the class numbers of real quadratic fields, Nagoya Math. J. 95 (1984), 125-135. https://doi.org/10.1017/S0027763000021036
  3. B. D. Beach, H. C. Williams, and C. R. Zarnke, Some computer results on units in quadratic and cubic fields, in Proceedings of the Twenty-Fifth Summer Meeting of the Canadian Mathematical Congress (Lakehead Univ., Thunder Bay, Ont., 1971), 609-648, Lakehead Univ., Thunder Bay, ON, 1971.
  4. L. Bernstein, Fundamental units and cycles in the period of real quadratic number fields. I, Pacific J. Math. 63 (1976), no. 1, 37-61. http://projecteuclid.org/euclid.pjm/1102867566 https://doi.org/10.2140/pjm.1976.63.37
  5. L. Bernstein, Fundamental units and cycles in the period of real quadratic number fields. II, Pacific J. Math. 63 (1976), no. 1, 63-78. http://projecteuclid.org/euclid.pjm/1102867567 https://doi.org/10.2140/pjm.1976.63.63
  6. D. Byeon and S. Lee, A note on units of real quadratic fields, Bull. Korean Math. Soc. 49 (2012), no. 4, 767-774. https://doi.org/10.4134/BKMS.2012.49.4.767
  7. D. Chakraborty and A. Saikia, On a conjecture of Mordell, Rocky Mountain J. Math. 49 (2019), no. 8, 2545-2556. https://doi.org/10.1216/RMJ-2019-49-8-2545
  8. C. Friesen, On continued fractions of given period, Proc. Amer. Math. Soc. 103 (1988), no. 1, 9-14. https://doi.org/10.2307/2047518
  9. R. Hashimoto, Ankeny-Artin-Chowla conjecture and continued fraction expansion, J. Number Theory 90 (2001), no. 1, 143-153. https://doi.org/10.1006/jnth.2001.2652
  10. J. Mc Laughlin, Multi-variable polynomial solutions to Pell's equation and fundamental units in real quadratic fields, Pacific J. Math. 210 (2003), no. 2, 335-349. https://doi.org/10.2140/pjm.2003.210.335
  11. R. A. Mollin, Quadratics, CRC Press Series on Discrete Mathematics and its Applications, CRC Press, Boca Raton, FL, 1996.
  12. L. J. Mordell, On a Pellian equation conjecture, Acta Arith. 6 (1960), 137-144. https://doi.org/10.4064/aa-6-2-137-144
  13. L. J. Mordell, On a Pellian equation conjecture. II, J. London Math. Soc. 36 (1961), 282-288. https://doi.org/10.1112/jlms/s1-36.1.282
  14. O. Ozer and F. K. Telci, On continued fractions of real quadratic fields with period six, Int. J. Contemp. Math. Sci. 6 (2011), no. 17-20, 833-840.
  15. O. Ozer, F. K. Telci, and H. I,scan, On some real quadratic fields with period 4, Int. J. Contemp. Math. Sci. 4 (2009), no. 25-28, 1389-1396.
  16. O. Perron, Die Lehre von den Kettenbruchen. Bd I. Elementare Kettenbruche, B. G. Teubner Verlagsgesellschaft, Stuttgart, 1954.
  17. K. Tomita, Explicit representation of fundamental units of some quadratic fields, Proc. Japan Acad. Ser. A Math. Sci. 71 (1995), no. 2, 41-43. http://projecteuclid.org/euclid.pja/1195510812 https://doi.org/10.3792/pjaa.71.41
  18. K. Tomita, Explicit representation of fundamental units of some real quadratic fields. II, J. Number Theory 63 (1997), no. 2, 275-285. https://doi.org/10.1006/jnth.1997.2088
  19. A. J. van der Poorten, H. J. J. te Riele, and H. C. Williams, Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000, Math. Comp. 70 (2001), no. 235, 1311-1328. https://doi.org/10.1090/S0025-5718-00-01234-5
  20. A. J. van der Poorten, H. J. J. te Riele, and H. C. Williams, Corrigenda and addition to: "Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000, Math. Comp. 72 (2003), no. 241, 521-523. https://doi.org/10.1090/S0025-5718-02-01527-2