On a relationship between the convergents of the nearest integer and regular continued fractions

Author:
William W. Adams

Journal:
Math. Comp. **33** (1979), 1321-1331

MSC:
Primary 10K10; Secondary 10K15

DOI:
https://doi.org/10.1090/S0025-5718-1979-0537978-9

MathSciNet review:
537978

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Abstract: In this paper we derive a relation concerning the speed of convergence of the nearest integer and regular continued fractions. If , denote the convergents of the nearest integer and regular continued fractions of an irrational number , then for all *n* there is a such that . It is shown that

**[1]**P. BILLINGSLEY,*Ergodic Theory and Information*, Wiley, New York, 1965. MR**0192027 (33:254)****[2]**A. KHINTCHINE,*Continued Fractions*, Univ. of Chicago Press, Chicago, 1964.**[3]**W. PHILIPP, "Some metrical theorems in number theory,"*Pacific J. Math.*, v. 29, 1967, pp. 109-127. MR**0205930 (34:5755)****[4]**D. SHANKS, Review of the UMT file: "Two related quadratic surds having continued fractions with exceptionally long periods,"*Math. Comp.*, v. 28, 1974, pp. 333-334. MR**0352049 (50:4537)****[5]**J. SCHOCKLEY,*Introduction to Number Theory*, Holt, Rinehart and Winston, New York, 1967. MR**0210649 (35:1535)****[6]**H. WILLIAMS & J. BROERE, "A computational technique for evaluating and the class number of a real quadratic field,"*Math. Comp.*, v. 30, 1976, pp. 887-893. MR**0414522 (54:2623)****[7]**H. WILLIAMS & P. BUHR, "Calculation of the regulator of by use of the nearest integer continued fraction algorithm,"*Math. Comp.*, v. 33, 1979, pp. 369-381. MR**514833 (80e:12003)****[8]**H. WILLIAMS, "Some results concerning the nearest integer continued fraction algorithm,"*J. Reine Angew. Math.*(To appear.)**[9]**G. J. RIEGER, "Über die mittlere Schrittanzahl bei Divisionsalgorithmen,"*Math. Nachr.*, v. 82, 1978, pp. 157-180. MR**0480366 (58:533)**

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DOI:
https://doi.org/10.1090/S0025-5718-1979-0537978-9

Article copyright:
© Copyright 1979
American Mathematical Society