Project Euler

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# [[Problem 71]]

## Ordered Fractions

Consider the fraction, $\dfrac n d$, where $n$ and $d$ are positive integers. If $n \lt d$ and $\operatorname{HCF}(n,d)=1$, it is called a reduced proper fraction.

If we list the set of reduced proper fractions for $d \le 8$ in ascending order of size, we get:

$$\frac 1 8, \frac 1 7, \frac 1 6, \frac 1 5, \frac 1 4, \frac 2 7, \frac 1 3, \frac 3 8, \mathbf{\frac 2 5}, \frac 3 7, \frac 1 2, \frac 4 7, \frac 3 5, \frac 5 8, \frac 2 3, \frac 5 7, \frac 3 4, \frac 4 5, \frac 5 6, \frac 6 7, \frac 7 8$$

It can be seen that $\dfrac 2 5$ is the fraction immediately to the left of $\dfrac 3 7$.

By listing the set of reduced proper fractions for $d \le 1\,000\,000$ in ascending order of size, find the numerator of the fraction immediately to the left of $\dfrac 3 7$.

## 順序分数

nとdを正の整数として, 分数 n/d を考えよう. n<d かつ HCF(n,d)=1 のとき, 真既約分数と呼ぶ.

d ≤ 8について既約分数を大きさ順に並べると, 以下を得る:

<div style="text-align:center">1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, <b>2/5</b> , 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8</div>

3/7のすぐ左の分数は2/5である.

d ≤ 1,000,000について真既約分数を大きさ順に並べたとき, 3/7のすぐ左の分数の分子を求めよ.

```scheme

(import (scheme base)

        (gauche base))

;;; ファレイ数列の性質から、3/7 の左隣は

;;; 3b - 7a = 1 を満たす a/b である

;;; b <= 10^6 を満たす中で最大の b を求め

;;; その時の a/b が答えになる

(define (a-b)

  (let loop ([b #e1e6])

    ; 7a = 3b - 1

    (let ([current (- (* b 3) 1)])

      (if (zero? (mod current 7))

        `(,(div current 7) ,b)

        (loop (- b 1))))))

(define answer-71 (car (a-b)))

(format #t "71: ~d~%" answer-71)

```