■平方根と連分数（その５７）
α=[a0:a1,a2,・・・]
λ=[an+1:an+2,an+3,・・・]+[0:an,an-1,・・・,a1]
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α→λ=[2;2,1,1,2,2,1,1,,,]+[0;1,1,2,2,1,1,2,2,,,]
→(9+√221)/10+(-9+√221)/10=√221/5
α→λ=[2;1,1,1,1,,,]+[0;2,1,1,1,,,]
→φ+1+1/(φ+1)=3
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x=[2:1,2,2,1,2,・・・],y=1/x
x=2+1/(1+1/(2+1/x))
x=2+1/(1+x/(2x+1))
x=2+(2x+1)/(3x+1)=(8x+3)/(3x+1)
3x^2-7x-3=0
x={7+(85)^1/2}/6
y=6/{7+(85)^1/2}={(85)^1/2-7}/6
λ=(85)^1/2/3=(9・9+4）/3
λ=(9・Q^2+4）^1/2/Q
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x=[2:1,1,2,2,1,1,2,・・・],y=1/x
x=2+1/(1+1/(1+1/(2+1/x))
x=2+1/(1+1/(1+x/(2x+1))
x=2+1/(1+(2x+1)/(3x+1))
x=2+(3x+1)/(5x+2)
x=(13x+5)/(5x+2)
5x^2-11x-5=0
x={11+(221)^1/2}/10
y=10/{11+(221)^1/2}={(221)^1/2-11}/10
λ=(221)^1/2/5=(9・25+4）^1/2/5にならない
λ=(9・Q^2-4）^1/2/Qになる
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
λ=(9・25-4）^1/2/5=(9・Q^2-4）^1/2/Qになっている
xをyで表す方が簡単である。
x=2+1/(2+y)=(2y+5)/(y+2)
λ=x+y=(2y+5)/(y+2)+y=(y^2+4y+5)/(y+2)＝√221/5
y^2+(4-λ)y+(5-2λ)=0
y={(λ-4)+{(4-λ)^2-4(5-2λ)}^1/2}/2
y={(λ-4)+{λ^2-4}^1/2}/2 ＝{√221/5-4+11/5}/2={√221-9}/10
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x=[2:1,1,1,2,2,1,1,1,2,・・・],y=1/x
x=2+1/(1+1/(1+1/(1+1/(2+1/x))
x=2+1/(1+1/(1+1/(1+x/(2x+1))
x=2+1/(1+1/(1+(2x+1)/(3x+1))
x=2+1/(1+(3x+1)/(5x+2)
x=2+(5x+2)/(8x+3)
x=(21x+8)/(8x+3)
8x^2-18x-8=0
x={9+(145)^1/2}/8
y=8/{9+(145)^1/2}={(145)^1/2-9}/8
λ=(145)^1/2/4=(9・16+4）^1/2/4
λ=(9・Q^2+4）^1/2/Q・・・になっていない
x={18+(580)^1/2}/16
ｙ={-18+(580)^1/2}/16
λ=(580)^1/2/8=(9・64+4）^1/2/4
λ=(9・Q^2+4）^1/2/Q・・・になっている
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α→λ=[2;1,1,1,1,2,2,1,1,,,]+[0;2,1,1,1,1,2,2,1,1,1,1,,,]
x=2+1/(1+1/(1+1/(1+1/(1+1/(2+1/x))
x=2+1/(1+1/(1+1/(1+1/(1+x/(2x+1))
x=2+1/(1+1/(1+1/(1+(2x+1)/(3x+1))
x=2+1/(1+1/(1+(3x+1)/(5x+2))
x=2+1/(1+(5x+2)/(8x+3))
x=2+(8x+3)/(13x+5)
x=(34x+13)/(13x+5)
13x^2-29x-13=0
x={29+(1517)^1/2}/26
y=1/x=26/{29+(1517)^1/2}={-29+(1517)^1/2}/26・・・両者は等しい
α→λ=[2;2,1,1,1,1,2,2,1,1,,,]+[0;1,1,1,1,2,2,1,1,1,1,,,]
x=2+1/(2+1/(1+1/(1+1/(1+1/(1+1/x))
x=2+1/(2+1/(1+1/(1+1/(1+x/(x+1))
x=2+1/(2+1/(1+1/(1+(x+1)/(2x+1))
x=2+1/(2+1/(1+(2x+1)/(3x+2))
x=2+1/(2+(3x+2)/(5x+3))
x=2+(5x+3)/(13x+8)=(31x+19)/(13x+8)
13x^2-23x-19=0
x={23+(1517)^1/2}/26
y=1/(1+1/(1+1/(1+1/(1+1/(2+1/(2+y))
y=1/(1+1/(1+1/(1+1/(1+(y+2)/(2y+5))
y=1/(1+1/(1+1/(1+(2y+5)/(3y+7))
y=1/(1+1/(1+(3y+7)/(5y+12))
y=1/(1+(5y+12)/(8y+19))
y=(8y+19)/(13y+31)
13y^2+23y-19=0
y={-23+(1517)^1/2}/26
λ=(1517)^1/2/13=(1517）^1/2/13
→(9・169-4)＾1/2/13
xをyで表す方が簡単である。
x=2+1/(2+y)=(2y+5)/(y+2)
λ=x+y=(2y+5)/(y+2)+y=(y^2+4y+5)/(y+2)＝√1517/13
y^2+(4-λ)y+(5-2λ)=0
y={(λ-4)+{(4-λ)^2-4(5-2λ)}^1/2}/2
y={(λ-4)+{λ^2-4}^1/2}/2 ＝{√1517/13-4+29/13}/2={√1517-23}/26
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α→λ=[2;2,1,1,1,1,1,1,2,2,1,1,,,]+[0;1,1,1,1,1,1,2,2,1,1,1,1,,,]
x=2+1/(2+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/x))
x=2+1/(2+1/(1+1/(1+1/(1+1/(1+1/(1+x/(x+1))
x=2+1/(2+1/(1+1/(1+1/(1+1/(1+(x+1)/(2x+1))
x=2+1/(2+1/(1+1/(1+1/(1+(2x+1)/(3x+2))
x=2+1/(2+1/(1+1/(1+(3x+2)/(5x+3))
x=2+1/(2+1/(1+(5x+3)/(8x+5))
x=2+1/(2+(8x+5)/(13x+8))
x=2+(13x+8)/(34x+21)
x=(81x+50)/(34x+21)
34x^2-60x-50=0
x={60+(10400)^1/2}/68
y=1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(2+1/(2+y))
y=1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+(y+2)/(2y+5))
y=1/(1+1/(1+1/(1+1/(1+1/(1+(2y+5)/(3y+7))
y=1/(1+1/(1+1/(1+1/(1+(3y+7)/(5y+12))
y=1/(1+1/(1+1/(1+(5y+12)/(8y+19))
y=1/(1+1/(1+(8y+19)/(13y+31))
y=1/(1+(13y+31)/(21y+50))
y=(21y+50)/(34y+81)
34y^2+60y-50=0
y={-60+(10400)^1/2}/68
λ=(10400)^1/2/34=
→(9・1156-4)＾1/2/34
xをyで表す方が簡単である。
x=2+1/(2+y)=(2y+5)/(y+2)
λ=x+y=(2y+5)/(y+2)+y=(y^2+4y+5)/(y+2)＝√10400/34
y^2+(4-λ)y+(5-2λ)=0
y={(λ-4)+{(4-λ)^2-4(5-2λ)}^1/2}/2
y={(λ-4)+{λ^2-4}^1/2}/2 ＝{√10400/34-4+76/34}/2={√10400-60}/68
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x=[2:1,1,1,1,1,2,2,1,1,1,1,1,2,・・・],y=1/x
x=2+1/(1+1/(1+1/(1+1/(1+1/(1+1/(2+1/x))
x=2+1/(1+1/(1+1/(1+1/(1+1/(1+x/(2x+1))
x=2+1/(1+1/(1+1/(1+1/(1+(2x+1)/(3x+1))
x=2+1/(1+1/(1+1/(1+(3x+1)/(5x+2))
x=2+1/(1+1/(1+(5x+2)/(8x+3))
x=2+1/(1+(8x+3)/(13x+5))
x=2+(13x+5)/(21x+8)
x=(55x+21)/(21x+8)
21x^2-47x-21=0
x={47+(3973)^1/2}/42
y=42/{-47+(3973)^1/2}={(3973)^1/2-47}/42
λ=(3973)^1/2/21=(9・441+4）^1/2/21
λ=(9・Q^2+4）^1/2/Q・・・なりたつ
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x=[2:1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,1,2,・・・],y=1/x
x=2+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(2+1/x))
x=2+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+x/(2x+1))
x=2+1/(1+1/(1+1/(1+1/(1+1/(1+1/(1+(2x+1)/(3x+1))
x=2+1/(1+1/(1+1/(1+1/(1+1/(1+(3x+1)/(5x+2))
x=2+1/(1+1/(1+1/(1+1/(1+(5x+2)/(8x+3))
x=2+1/(1+1/(1+1/(1+(8x+3)/(13x+5))
x=2+1/(1+1/(1+(13x+5)/(21x+8))
x=2+1/(1+(21x+8)/(34x+13))
x=2+(34x+13)/(55x+21)
x=(144x+55)/(55x+21)
55x^2-123x-55=0
x={123+(27229)^1/2}/110
y=110/{-123+(27229)^1/2}={(27229)^1/2-123}/110
λ=(27229)^1/2/55=(9・3025+4）^1/2/55
λ=(9・Q^2+4）^1/2/Q・・・なりたつ
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これでフィボナッチ数になることははっきりした
n=0:x=2+(2x+1)/(3x+1)=(8x+3)/(3x+1)=(F6x+F4)/((F4x+F2)
n=1:x=(21x+8)/(8x+3)=(F8x+F6)/(F6x+F4)
n=2:x=(55x+21)/(21x+8)=(F10x+F8)/(F8x+F6)
3=3:x=(144x+55)/(55x+21)=(F12x+F10)/(F10x+F8)
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ペル数が出現するλ=√7565/29について
9・29・29-4＝7565
ax^2+bx+c=0
a=29,b^2-4ac=7565
b^2-116c=7565
c=41, b=111
c=61, b=121
c=949, b=343
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cは負が考えられる
c=-41, b=53
c=-65, b=5
c=-41, b=53
29x^2-53x-41=0
x={53+(7565)^1/2}/58/58>2・・・???
y=1/x=58/{53+(7565)^1/2={-53+(7565)^1/2/82・・・分母が異なっている
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c=-65, b=5
29x^2-5x-65=0
29x^2-5x-65=0
ｘ＝{5+（7565）^1/2}/58>1???
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c=41, b=111
29x^2-111x-41=0
ｘ＝{111+（7565）^1/2}/58>3
y=1/x=58/{111+（7565）^1/2}={111-（7565）^1/2}/82・・・分母が異なっている
y=1/(1+1/(1+1/x))=1/(1+1/(x+1)/x)=1/(1+x/(x+1))=(x+1)/(2x+1)
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xをyで表す方が簡単である。
x=2+1/(2+y)=(2y+5)/(y+2)
λ=x+y=(2y+5)/(y+2)+y=(y^2+4y+5)/(y+2)＝√7565/29
y^2+(4-λ)y+(5-2λ)=0
y={(λ-4)+{(4-λ)^2-4(5-2λ)}^1/2}/2
y={(λ-4)+{λ^2-4}^1/2}/2・・・ 根号の中が簡単にならない。したがって、x,yの形が異なっている
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x=[2:2,1,2,2,1,・・・]
y=[0,1,2,2,1,2,2,・・・]の場合
x=2+1/(2+1/(1+1/x)
x=2+1/(2+x/(x+1))
x=2+(x+1)/(3x+2)=(7x+5)/(2x+2)
2x^2-5x-5=0
x={5+(65)^1/2}/4・・・該当なし
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