# Length of continued fractions in principal quadratic fields

Acta Arithmetica (1998)

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

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topGuillaume 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

top- [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] S. Chowla and S. S. Pillai, Periodic simple continued fraction, J. London Math. Soc. 6 (1931), 85-89. Zbl0001.32601
- [3] H. Cohen, Multiplication par un entier d'une fraction continue périodique, Acta Arith. 26 (1974), 129-148. Zbl0273.10031
- [4] D. A. Cox, Primes of the Form x² + ny², Wiley, 1989.
- [5] R. Descombes, Eléments de théorie des nombres, Presses Univ. France, 1986.
- [6] G. Grisel, Sur la longueur de la fraction continue de ${\alpha}^{n}$, Acta Arith. 74 (1996), 161-176.
- [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] R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer, 1994. Zbl0805.11001
- [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] L. J. Mordell, On a Pellian equation conjecture, Acta Arith. 6 (1960), 137-144. Zbl0093.04305

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