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Problem 57¶

Square Root Convergents¶

It is possible to show that the square root of two can be expressed as an infinite continued fraction.

$\sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}}$

By expanding this for the first four iterations, we get:

$1 + \frac 1 2 = \frac 32 = 1.5$
$1 + \frac 1 {2 + \frac 1 2} = \frac 7 5 = 1.4$
$1 + \frac 1 {2 + \frac 1 {2+\frac 1 2}} = \frac {17}{12} = 1.41666 \dots$
$1 + \frac 1 {2 + \frac 1 {2+\frac 1 {2+\frac 1 2}}} = \frac {41}{29} = 1.41379 \dots$

The next three expansions are $\frac {99}{70}$, $\frac {239}{169}$, and $\frac {577}{408}$, but the eighth expansion, $\frac {1393}{985}$, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.

In the first one-thousand expansions, how many fractions contain a numerator with more digits than the denominator?

平方根の近似分数¶

2の平方根は無限に続く連分数で表すことができる.

最初の4回の繰り返しを展開すると以下が得られる.

1 + 1/2 = 3/2 = 1.5
1 + 1/(2 + 1/2) = 7/5 = 1.4
1 + 1/(2 + 1/(2 + 1/2)) = 17/12 = 1.41666...
1 + 1/(2 + 1/(2 + 1/(2 + 1/2))) = 41/29 = 1.41379...

次の3つの項は99/70, 239/169, 577/408である. 第8項は1393/985である. これは分子の桁数が分母の桁数を超える最初の例である.

最初の1000項を考えたとき, 分子の桁数が分母の桁数を超える項はいくつあるか?

;;;
;;; Gaucheコード（なぜか不正解になる…）
;;;
(import (scheme base)
        (gauche base)
        (scheme inexact))

(define (digit-num num)
  (assume (exact-integer? num))
  (assume (not (negative? num)))
  (if (zero? num)
    1
    (+ (floor->exact (log num 10))
       1)))

(define continued-fraction
  (let loop ([index 1] [current 1] [result '()])
    (if (< 1000 index)
      (reverse result)
      (let ([next (+ 1
                     (/ 1 
                        (+ current 1)))])
        (loop (+ index 1)
              next
              (cons next result))))))

(define target-fraction
  (filter (^r
            (< (digit-num (denominator r))
               (digit-num (numerator r))))
          continued-fraction))

(define answer-57
  (length target-fraction))

(format #t "57: ~d~%" answer-57)

;;;
;;; Chez Schemeコード（正解）
;;;
#!r6rs
#!chezscheme

(import (chezscheme))

(define (digit-num num)
  (assert (exact? num))
  (assert (integer? num))
  (assert (not (negative? num)))
  (if (zero? num)
    1
    (+ (exact (floor (log num 10)))
       1)))

(define continued-fraction
  (let loop ([index 1] [current 1] [result '()])
    (if (< 1000 index)
      (reverse result)
      (let ([next (+ 1
                     (/ 1 
                        (+ current 1)))])
        (loop (+ index 1)
              next
              (cons next result))))))

(define target-fraction
  (filter (lambda (r)
            (< (digit-num (denominator r))
               (digit-num (numerator r))))
          continued-fraction))

(define answer-57
  (length target-fraction))

(printf "57: ~d~%" answer-57)