フェルマーの最終定理の証明 (772ﾚｽ)
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抽出解除 必死ﾁｪｯｶｰ(本家) (べ) 自ID ﾚｽ栞 あぼーん

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673: 08/20(水)10:07 ID:kS5YreVJ(1/9) AAS
1/m+1/n=3/77 (m>n)を満たす自然数の組(m,n)
1/m+1/n=(m+n)/mn=3/77 77(m+n)=3mn
3mn-77m-77n=0
9mn-231m-231n=0
(3m-77)(3n-77)-77^2=0
(3m-77)(3n-77)=77^2=7^2･11^2
(7^2･11^2,1), (7･11^2,7),(7^2･11,11),(11^2･7,7),(11^2,7^2 )
?(7^2･11^2,1)のとき
3m-77=7^2･11^2 3m=7^2･11^2+77=6006 ∴m=2002
3n-77=1 3n=78 ∴n=26
省9

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674: 08/20(水)10:09 ID:kS5YreVJ(2/9) AAS
e^(-1.822r)( 0.223 + 0.223e^r + 0.223e^(2r) + 0.331e^(3r) ) = 1
0.223e^(-1.822r) + 0.223e^(-0.822r) + 0.223e^(0.178r) + 0.331e^(1.178r) = 1
f(r) = 0.223e^(-1.822r) + 0.223e^(-0.822r) + 0.223e^(0.178r) + 0.331e^(1.178r)

f'(r) = -0.406306e^(-1.822r) - 0.183306e^(-0.822r) + 0.039694e^(0.178r) + 0.389918e^(1.178r)
f'(0) = -0.16
f(r) は下に凸で f(0) = 1,f(1) ≒ 1.5 なので 0 < r < 1 の範囲に r = 0 以外の解がある。
a[0] = 1
a[n]-{0.223e^(-1.822a[n])+0.223e^(-0.822a[n])+0.223e^(0.178a[n])+0.331e^(1.178a[n])-1}
a[n+1] = ───────────────────────────────────────────
{-0.406306e^(-1.822a[n])-0.183306e^(-0.822a[n])+0.039694e^(0.178a[n])+0.389918e^(1.178a[n])}
省8

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675: 08/20(水)10:10 ID:kS5YreVJ(3/9) AAS
{! 123456789 <─-i
101110011 1*2^8+1*2^7+･････+1*2^0 i
x := 2*x+v 2*0+1 1
2*(2*0+1)+0 2
2*(2*(2*0+1)+0)+1 3
2*(2*(2*(2*0+1)+0)+1)+1 4
2*(2*(2*(2*(2*0+1)+0)+1)+1)+1 5
2*(2*(2*(2*(2*(2*0+1)+0)+1)+1)+1)+0 6
2*(2*(2*(2*(2*(2*(2*0+1)+0)+1)+1)+1)+0)+0 7
2*(2*(2*(2*(2*(2*(2*(2*0+1)+0)+1)+1)+1)+0)+0)+1 8
省15

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676: 08/20(水)10:17 ID:kS5YreVJ(4/9) AAS
∫[a→b]dx/√((x-a)(x-b))
=∫[a→b]dx/√{-x^2+(a+b)x-ab}
=∫[a→b]dx/√{(1/4)(a+b)^2-ab-{x-(a+b)/2}^2}
=∫[a→b]dx/√{(1/4)(a-b)^2-{x-(a+b)/2}^2}
= {2/(b-a)}∫[a→b]dx/√{1-{2{x-(a+b)/2}/(b-a)}^2}
= [arcsin{2{x-(a+b)/2}/(b-a)}][a→b]
=π

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677: 08/20(水)10:23 ID:kS5YreVJ(5/9) AAS
f(z)=1/(1-z) z=i で展開
?@) |z-i|<√2
(1-i)(1-(z-i)/(1-i))=1-i+(1-i) (z-i)/(1-i)
1/(1-z)=1/(1-i-(z-i) )=1/(1-i)?1/(1-(z-i)/(1-i))
=1/(1-i) (1+((z-i)/(1-i))+((z-i)/(1-i))^2+((z-i)/(1-i))^3+?)
=(z-i)^0/(1-i)+(z-i)^1/(1-i)^2 +(z-i)^2/(1-i)^3 +?
=((1+i) (z-i)^0)/2+((1+i)^2 (z-i)^1)/2^2 +((1+i)^3 (z-i)^2)/2^3 +?
=?[n=0→∞]((1+i)/2)^(n+1) (z-i)^n
※(1 )/(1-i)^2 =(1/(1-i))(1/(1-i))=(1+i)/((1-i)(1+i))((1+i)/(1-i)(1+i)) =(1+i)^2/2^2

?A) |z-i|>√2の場合
省12

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681: 08/20(水)18:15 ID:kS5YreVJ(6/9) AAS
y''(t) - 3y'(t) + 2y(t) = e^(-t) ･･･････?（初期条件）y(0) = 1/6, y'(0) = 5/6

L[y''(t)] = s^2Y(s) - sy(0) - y'(0) = s^2Y(s) - s/6 - 5/6
L[3y'(t)] = 3( sY(s) - y(0) ) = 3sY(s) - 1/2
L[2y(t)] = 2Y(s)
L[e^(-t)] = 1/(s + 1)

s^2Y(s) - s/6 - 5/6 - (3sY(s) -1/2) + 2Y(s) = 1/(s+1)
Y(s)(s^2 - 3s + 2) - s/6 -1/3 = 1/(s+1)
s 1 1
Y(s)(s-1)(s-2) = ─ + ─ + ──
6 3 s+1
省10

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682: 08/20(水)18:16 ID:kS5YreVJ(7/9) AAS
M(θ)=E[e^θX ]=∫_(-∞)^∞??e^θx f(x)dx?
M(θ)=E[e^θX ]=1/(√2π σ) ∫_(-∞)^∞??e^θx e^(-(x-μ)^2/(2σ^2 )) ? dx=1/(√2π σ) ∫_(-∞)^∞?e^(θx-(x-μ)^2/(2σ^2 )) dx
θx-(x-μ)^2/(2σ^2 )=1/(2σ^2 ) (2σ^2 θx-(x-μ)^2 )=-1/(2σ^2 ) (? (x-μ)?^2-2σ^2 θx )
=-1/(2σ^2 ) (? x?^2+μ^2-2μx-2σ^2 θx )
=-1/(2σ^2 ) (? x?^2-2(μ+σ^2 θ)x+μ^2 )
=-1/(2σ^2 ) ((x-(μ+σ^2 θ))^2-(μ+σ^2 θ)^2+μ^2 )
=-1/(2σ^2 ) ((x-(μ+σ^2 θ))^2-(μ^2+2μσ^2 θ+σ^4 θ^2 )+μ^2 )
=-1/(2σ^2 ) ((x-(μ+σ^2 θ))^2-(2μσ^2 θ+σ^4 θ^2 ) )
=-(x-(μ+σ^2 θ))^2/(2σ^2 )+μθ+(σ^2 θ^2)/2
M(θ)=1/(√2π σ) ∫_(-∞)^∞?e^(θx-(x-μ)^2/(2σ^2 )) dx
省5

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683: 08/20(水)18:25 ID:kS5YreVJ(8/9) AAS
y''(t) - 3y'(t) + 2y(t) = e^(-t) ･･･････?（初期条件）y(0) = 1/6, y'(0) = 5/6
L[y''(t)] = s^2Y(s) - sy(0) - y'(0) = s^2Y(s) - s/6 - 5/6
L[3y'(t)] = 3( sY(s) - y(0) ) = 3sY(s) - 1/2
L[2y(t)] = 2Y(s)
L[e^(-t)] = 1/(s + 1)

s^2Y(s) - s/6 - 5/6 - (3sY(s) -1/2) + 2Y(s) = 1/(s+1)
Y(s)(s^2 - 3s + 2) - s/6 -1/3 = 1/(s+1)
Y(s)(s-1)(s-2) = s/6+1/3+1/(s+1) = (s(s+1)+2(s+1)+6)/6(s+1) = (s^2 + 3s + 8)/6(s+1)
Y(s) = (s^2 + 3s + 8)/6(s+1)(s-1)(s-2) = A/(s+1) + B/(s-1) + C/(s-2)
s^2 + 3s + 8 = 6( A(s-1)(s-2) + B(s+1)(s-2) + C(s+1)(s-1) )
省5

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684: 08/20(水)18:25 ID:kS5YreVJ(9/9) AAS
y''(t) - 3y'(t) + 2y(t) = e^(-t) ･･･････?（初期条件）y(0) = 1/6, y'(0) = 5/6
k^2 -3k + 2 = (k-1)(k-2) = 0 k = 1, 2
y''(t) - 3'y(t) + 2y(t) = 0
y0 = C1e^t + C2e^(2t)
v(t) = 1/(D-1)(D-2)*e^(-t)
= 1/(D-2)*e^(-t) - 1/(D-1)*e^(-t)
= (-1/3)e^(-t) + (1/2)e^(-t) = (1/6)e^(-t)
y(t) = C1e^t + C2e^(2t) + (1/6)e^(-t)
y(0) = C1 + C2 + 1/6 = 1/6
C1 + C2 = 0 …… ?
省6

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