Length of continued fractions in principal quadratic fields

• Autorzy: Guillaume Grisel
• Języki publikacji: EN

Abstrakty

EN

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].

Kategorie tematyczne

• 11A55: Continued fractions
• 11J70: Continued fractions and generalizations

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1998
• Tom: 85
• Numer: 1
• Strony: 35-49

Daty

wydano

1998

otrzymano

1997-01-10

poprawiono

1997-11-25

Twórcy

autor

Guillaume Grisel

• Département de Mathématiques, Université de Caen, Esplanade de la Paix, 14032 Caen Cedex, France

Bibliografia

• [1] N. C. Ankeny, E. Artin and S. Chowla, The class number of real quadratic number fields, Ann. of Math. 56 (1952), 479-493.
• [2] S. Chowla and S. S. Pillai, Periodic simple continued fraction, J. London Math. Soc. 6 (1931), 85-89.
• [3] H. Cohen, Multiplication par un entier d'une fraction continue périodique, Acta Arith. 26 (1974), 129-148.
• [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 $α^n$, 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.
• [8] R. K. Guy, Unsolved Problems in Number Theory, 2nd ed., Springer, 1994.
• [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.