Length of continued fractions in principal quadratic fields

Guillaume Grisel

Acta Arithmetica (1998)

  • Volume: 85, Issue: 1, page 35-49
  • ISSN: 0065-1036

Abstract

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Let d ≥ 2 be a square-free integer and for all n ≥ 0, let l ( ( √ d ) 2 n + 1 ) be the length of the continued fraction expansion of ( √ d ) 2 n + 1 . If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that C ₁ ( √ d ) 2 n + 1 ≥ l ( ( √ d ) 2 n + 1 ) ≥ C ₂ ( √ d ) 2 n + 1 for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].

How to cite

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Guillaume Grisel. "Length of continued fractions in principal quadratic fields." Acta Arithmetica 85.1 (1998): 35-49. <http://eudml.org/doc/207153>.

@article{GuillaumeGrisel1998,
abstract = {Let d ≥ 2 be a square-free integer and for all n ≥ 0, let $l((√d)^\{2n+1\})$ be the length of the continued fraction expansion of $(√d)^\{2n+1\}$. If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that $C₁(√d)^\{2n+1\} ≥ l((√d)^\{2n+1\}) ≥ C₂(√d)^\{2n+1\}$ for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].},
author = {Guillaume Grisel},
journal = {Acta Arithmetica},
keywords = {principal quadratic fields; length of period; continued fraction expansion},
language = {eng},
number = {1},
pages = {35-49},
title = {Length of continued fractions in principal quadratic fields},
url = {http://eudml.org/doc/207153},
volume = {85},
year = {1998},
}

TY - JOUR
AU - Guillaume Grisel
TI - Length of continued fractions in principal quadratic fields
JO - Acta Arithmetica
PY - 1998
VL - 85
IS - 1
SP - 35
EP - 49
AB - Let d ≥ 2 be a square-free integer and for all n ≥ 0, let $l((√d)^{2n+1})$ be the length of the continued fraction expansion of $(√d)^{2n+1}$. If ℚ(√d) is a principal quadratic field, then under a condition on the fundamental unit of ℤ[√d] we prove that there exist constants C₁ and C₂ such that $C₁(√d)^{2n+1} ≥ l((√d)^{2n+1}) ≥ C₂(√d)^{2n+1}$ for all large n. This is a generalization of a theorem of S. Chowla and S. S. Pillai [2] and an improvement in a particular case of a theorem of [6].
LA - eng
KW - principal quadratic fields; length of period; continued fraction expansion
UR - http://eudml.org/doc/207153
ER -

References

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  1. [1] N. C. Ankeny, E. Artin and S. Chowla, The class number of real quadratic number fields, Ann. of Math. 56 (1952), 479-493. Zbl0049.30605
  2. [2] S. Chowla and S. S. Pillai, Periodic simple continued fraction, J. London Math. Soc. 6 (1931), 85-89. Zbl0001.32601
  3. [3] H. Cohen, Multiplication par un entier d'une fraction continue périodique, Acta Arith. 26 (1974), 129-148. Zbl0273.10031
  4. [4] D. A. Cox, Primes of the Form x² + ny², Wiley, 1989. 
  5. [5] R. Descombes, Eléments de théorie des nombres, Presses Univ. France, 1986. 
  6. [6] G. Grisel, Sur la longueur de la fraction continue de α n , Acta Arith. 74 (1996), 161-176. 
  7. [7] G. Grisel, Length of the continued fraction of the powers of a rational fraction, J. Number Theory 62 (1997), 322-337. Zbl0878.11028
  8. [8] R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer, 1994. Zbl0805.11001
  9. [9] M. Mendès France, The depth of a rational number, in: Topics in Number Theory (Debrecen, 1974), Colloq. Math. Soc. János Bolyai 13, North-Holland, 1976, 183-194. 
  10. [10] L. J. Mordell, On a Pellian equation conjecture, Acta Arith. 6 (1960), 137-144. Zbl0093.04305

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