α=[a0:a1,a2,・・・]

λ=[an+1:an+2,an+3,・・・]+[0:an,an-1,・・・,a1]

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α=Φのとき

λ＞[1:1,1,1,・・・]+[0:1,1,1,・・・,1]→φ+φ-1＝√5

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α=1+√2のとき

λ＞[2:2,2,2,・・・]+[0:2,2,2,・・・,2]→1+√2+√2-1＝√8

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α=(3+√13)/2のとき

λ＞[3:3,3,3,・・・]+[0:3,3,3,・・・,3]→(3+√13)/2+(-3+√13)/2＝√13

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α=(9+√221)/10=[2:2,1,1,2,2,1,1,・・・]のとき

λ＞[2:2,1,1,2,2,1,1,・・・]+[0:1,1,2,2,1,1,2,2,・・・,2]→(9+√221)/10+(-9+√221)/10＝√221/5

x=[2:2,1,1,2,2,1,1,・・・]

y=[0:1,1,2,2,1,1,2,2,・・・]=1/(1+1/(1+1/x))=1/(1+1/(x+1)/x)=1/(1+x/(x+1))=(x+1)/(2x+1)=(-9+√221)/10

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α=(√3)=[1:1,2,1,2,1,2,・・・]のとき

λ＞[2:1,2,1,2,・・・]+[0:1,2,1,2,・・・,1]→1+(√3)-1+√3＝√12

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α=(9√3+65)/22

(9√3+65)/22-3=(9√3-1)/22

22/(9√3-1)=(9√3+1)/11

(9√3+1)/11-1=(9√3-10)/11

11/(9√3-10)=(9√3+10)/13

(9√3+10)/13-1=(9√3-3)/13

13/(9√3-3)=(9√3+3)/18

(9√3+3)/18-1=(9√3-15)/18

18/(9√3-15)=(9√3+15)

(9√3+15)-30=(9√3-15)

1/(9√3-15)=(9√3+15)/18

(9√3+15)/18-1=(9√3-3)/18

18/(9√3-3)=(9√3+3)/13

(9√3+3)/13-1=(9√3-10)/13***

13/(9√3-10)=(9√3+10)/11

(9√3+10)/11-2=(9√3-12)/11

11/(9√3-12)=(9√3+12)/9

(9√3+12)/9-3=(9√3-15)/9

9/(9√3-15)=(9√3+15)/2

(9√3+15)/2-15=(9√3-15)/2

2/(9√3-15)=(9√3+15)/9

(9√3+15)/9-3=(9√3-12)/9

9/(9√3-12)=(9√3+12)/11

(9√3+12)/11-2=(9√3-10)/11

11/(9√3-10)=(9√3-10)/13***

λ=(9√3+65)/22=[3:1,1,1,30,1,1,2,3,15,3,2,2,3,15,3,2,2,3,15,3,2,・・・]誤り発見

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λ＞[2:3,15,3,2・・・]+[0:2,3,15,3,2,・・・]

x=[2:3,15,3,2・・・]

= 2+1/(3+1/(15+1/(3+1/(2+1/x)))

= 2+1/(3+1/(15+1/(3+x/(2x+1))

= 2+1/(3+1/(15+(2x+1)/(7x+3))

= 2+1/(3+(7x+3)/(107x+46))

=2+(107x+46)/(328x+141)

x=(763x+328)/(328x+141)

328x^2-622x-328=0

x={311+(311^2+328^2)^1/2}/328={311+452}/328=2.32

y=1/x=0.30

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λ＞[3:15,3,2,2・・・]+[0:2,2,3,15,3,2,・・・]

x=[3:15,3,2,2・・・]

= 3+1/(15+1/(3+1/(2+1/(2+1/x)))

= 3+1/(15+1/(3+1/(2+x/(2x+1))

= 3+1/(15+1/(3+(2x+1)/(5x+2))

= 3+1/(15+(5x+2)/(17x+7))

=3+(17x+7)/(260x+107)

x=(797x+328)/(260x+107)

260x^2-690x-328=0

x={345+(345^2+260・328)^1/2}/260={345+452}/260=3.06

y=1/(2+1/(2+1/x))=1/(2+x/(2x+1))=(2x+1)/(5x+2)=7.12/17.3=0.41

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3+1/1＞[3:1,・・・]＞3+1/2

3+1/2＞[3:2,・・・]＞3+1/3

3+1/3＞[3:2,・・・]＞3+1/4

1/1＞[0:1,・・・]＞1/2

1/2＞[0:2,・・・]＞1/3

1/3＞[0:3,・・・]＞1/4

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λ＞[3:2,2,3,15,・・・]+[0:2,2,3,15,3,2,・・・]

x=[3:2,2,3,15,・・・]

= 3+1/(2+1/(2+1/(3+1/(15+1/x)))

= 3+1/(2+1/(2+1/(3+x/(15x+1))

= 3+1/(2+1/(2+(15x+1)/(46x+3))

= 3+1/(2+(46x+3)/(107x+7))

=3+(107x+7)/(260x+17)

x=(887x+58)/(260x+17)

260x^2-870x-58=0

x={435+(435^2+260・58)^1/2}/260={435+452}/260=3.41

y=1/(2+1/(2+1/x))=1/(2+x/(2x+1))=(2x+1)/(5x+2)=7.82/19.0=0.41

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y=[0:15,3,2,1,1,30,1,1,1]

1/15＞y＞1/16

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