α⋍1/2+δ⒤ | β⋍1/2-δ⒤とおく | ||||||||||
(x-α)(x-β)⋍ | x^2-(α+β)x+αβ | ||||||||||
∵α+β⋍1、αβ⋍1/4+δ^2 | |||||||||||
⋍x^2-x+1/4+δ^2 | |||||||||||
⋍(x-1/2)^2+δ^2 | s=1+ℓとおくとルジャンドル | ||||||||||
①オメガωとルジャンドル | |||||||||||
ω^2+ω+1 | ⋍0 | ||||||||||
楕円曲 | y^2⋍ | x^3+Px+Q | x(x^2+δ^2)+Q | ||||||||
不変量 | j⋍ | 2^83^3P^3 | /( | 4P^3+27Q^2) | Q⋍0のときの不変量j | ||||||
δ^2 | -δ | 文献値 | 2^83^3P^3 | / | 4P^3+27Q^2) | j(一定) | |||||
∴11√ | 3 | 19.05256 | 5 | ⋍ | 14.05256 | ~ | 14.13472 | 6912 | / | 4 | 1728 |
∴11√ | 5 | 24.59675 | 4 | ⋍ | 21.09675 | ~ | 21.02203 | 55296 | / | 32 | 1728 |
∴11√ | 7 | 29.10326 | 4 | ⋍ | 25.10326 | ~ | 25.01085 | 186624 | / | 108 | 1728 |
∴11√ | 9 | 33 | 3 | ⋍ | 30.5 | ~ | 30.42487 | 442368 | / | 256 | 1728 |
∴11√ | 11 | 36.48287 | 4 | ⋍ | 32.98287 | ~ | 32.93506 | 864000 | / | 500 | 1728 |
∴11√ | 13 | 39.66106 | 2 | ⋍ | 37.66106 | ~ | 37.58617 | 1492992 | / | 864 | 1728 |
∴11√ | 15 | 42.60282 | 2 | ⋍ | 40.60282 | ~ | 40.91871 | 2370816 | / | 1372 | 1728 |
∴11√ | 17 | 45.35416 | 2 | ⋍ | 43.35416 | ~ | 43.32707 | 3538944 | / | 2048 | 1728 |
∴11√ | 19 | 47.94789 | 0 | ⋍ | 47.94789 | ~ | 48.00515 | 5038848 | / | 2916 | 1728 |
∴11√ | 21 | 50.40833 | 1 | ⋍ | 49.40833 | ~ | 49.77383 | 6912000 | / | 4000 | 1728 |
∴11√ | 23 | 52.75415 | 0 | ⋍ | 52.75415 | ~ | 52.97 | 9199872 | / | 5324 | 1728 |
∴11√ | 25 | 55 | 0 | ⋍ | 55 | ~ | 56.446 | 11943936 | / | 6912 | 1728 |
∴11√ | 27 | 57.15768 | 1 | ⋍ | 56.65768 | ~ | 56.446 | 15185664 | / | 8788 | 1728 |
∴11√ | 29 | 59.23681 | 0 | ⋍ | 59.23681 | ~ | 59.347 | 18966528 | / | 10976 | 1728 |
∴11√ | 31 | 61.24541 | 1 | ⋍ | 60.74541 | ~ | 60.831 | 23328000 | / | 13500 | 1728 |
∴11√ | 33 | 63.19019 | 0 | ⋍ | 63.19019 | ~ | 65.112 | 28311552 | / | 16384 | 1728 |
⓪ | 三角関数の合成関数がフックの法則を満たす. | ||||||
① | {1/φ^2,1/φ,φ,φ^2}が4次元を示すこと。 | ||||||
アレクサンダー | |||||||
C | t^2-t+1 | (t-ω)(t+ω) | |||||
O | t^2-3t+1 | (t-φ^2)(t-1/φ^2) | |||||
… | 2t^2-5t+2 | (t-0.5)(tー2) | |||||
n | ) | φ^n | ∓ | 1/φ^n | |||
1 | ) | 1.618 | ー | 0.618 | ⋍ | 1 | |
2 | ) | 2.618 | + | 0.381924 | ⋍ | 3 | |
3 | ) | 4.235801 | ー | 0.145866 | ⋍ | 4 | |
4 | ) | 17.943573 | + | 0.021277 | ⋍ | 18 |
①gμν⋍η-δ⋍11√(2k+1)-δ | |||||||
η | δ | 文献値 | |||||
11√ | 3 | 19.053 | - | 5 | ⋍ | 14.05256 | 14.13472 |
11√ | 5 | 24.597 | - | 3.5 | ⋍ | 21.09675 | 21.02203 |
11√ | 7 | 29.103 | - | 4 | ⋍ | 25.10326 | 25.01085 |
11√ | 9 | 33 | - | 2.5 | ⋍ | 30.5 | 30.42487 |
11√ | 11 | 36.483 | - | 3.5 | ⋍ | 32.98287 | 32.93506 |
11√ | 13 | 39.661 | - | 2 | ⋍ | 37.66106 | 37.58617 |
11√ | 15 | 42.603 | - | 2 | ⋍ | 40.60282 | 40.91871 |
11√ | 17 | 45.354 | - | 2 | ⋍ | 43.35416 | 43.32707 |
11√ | 19 | 47.948 | - | 0 | ⋍ | 47.94789 | 48.00515 |
11√ | 21 | 50.408 | - | 1 | ⋍ | 49.40833 | 49.77383 |
11√ | 23 | 52.754 | - | 0 | ⋍ | 52.75415 | 52.97 |
11√ | 25 | 55 | - | 0 | ⋍ | 55 | 56.446 |
11√ | 27 | 57.158 | - | 0.5 | ⋍ | 56.65768 | 56.446 |
11√ | 29 | 59.237 | - | 0 | ⋍ | 59.23681 | 59.347 |
11√ | 31 | 61.245 | - | 0.5 | ⋍ | 60.74541 | 60.831 |
11√ | 33 | 63.19 | - | 0 | ⋍ | 63.19019 | 65.112 |
α⋍1/2+δ⒤、 | β⋍1/2-δ⒤とおく | ||||||
(x-α)(x-β)⋍ | x^2-(α+β)x+αβ | ||||||
∵α+β⋍1、αβ⋍1/4+δ^2 | |||||||
⋍x^2-x+1/4+δ^2 | |||||||
⋍(x-1/2)^2+δ^2 | |||||||
(s)ー2-4-6ー8ー10… | |||||||
(1-s)3,5,7,9.11,… | |||||||
sと1-sを入れ替えても成立,s=1+ℓとおくと | |||||||
∴s(1-s)=ーℓ(ℓ+1)=ーα(α+1)ルジャンンドルの関係式使った | |||||||
x(x+1)+121[(2k+1.25)-δ]⋍0 | ⋍19.05-5⋍14.05 | ||||||
x^2+x+3.25-δ⋍0 ∴x⋍(-0.5)±[11√3-δ]⒤ | |||||||
p⋍(1/2)±11⒤[√(1-s)-(q/11)] | |||||||
文献値 | 文/11 | ( | s | ) | +s/11 | ⋍√2k+1 | ⋍2k+1 |
14.13472 | 1.28497454545455 | ( | 5 | ) | 0.45454545454546 | 1.73952 | 3.026 |
21.02203 | 1.91109363636364 | ( | 4 | ) | 0.36363636363636 | 2.27473 | 5.174 |
25.01085 | 2.27371363636364 | ( | 4 | ) | 0.36363636363636 | 2.63735 | 6.956 |
30.42487 | 2.76589727272727 | ( | 6 | ) | 0.54545454545455 | 3.31135 | 10.965 |
32.93506 | 2.99409636363636 | ( | 7 | ) | 0.63636363636364 | 3.63046 | 13.180 |
37.58617 | 3.41692454545455 | ( | 5 | ) | 0.45454545454546 | 3.87147 | 14.988 |
40.91871 | 3.71988272727273 | ( | 5 | ) | 0.45454545454546 | 4.17443 | 17.426 |
43.32707 | 3.93882454545455 | ( | 5 | ) | 0.45454545454546 | 4.39337 | 19.302 |
48.00515 | 4.36410454545455 | ( | 5 | ) | 0.45454545454546 | 4.81865 | 23.219 |
49.77383 | 4.52489363636364 | ( | 4 | ) | 0.36363636363636 | 4.88853 | 23.898 |
52.97 | 4.81545454545455 | ( | 5 | ) | 0.45454545454546 | 5.27000 | 27.773 |
56.446 | 5.13145454545455 | ( | 5 | ) | 0.45454545454546 | 5.58600 | 31.203 |
59.347 | 5.39518181818182 | ( | 2 | ) | 0.18181818181818 | 5.57700 | 31.103 |
60.831 | 5.53009090909091 | ( | 1 | ) | 0.09090909090909 | 5.62100 | 31.596 |
65.112 | 5.91927272727273 | ( | 3 | ) | 0.27272727272727 | 6.19200 | 38.341 |
75.8 | 6.89090909090909 | ( | 10 | ) | 0.90909090909091 | 7.80000 | 60.840 |
(s)ー2-4-6ー8ー10… | |||||||
(1-s)3,5,7,9.11,… | |||||||
sと1-sを入れ替えても成立,s=1+ℓとおく | |||||||
∴s(1-s)=ーℓ(ℓ+1)=ーα(α+1)ルジャンンドルの関係式使った | |||||||
x(x+1)+121[(2k+1.25)-δ]⋍0 | ⋍19.05-5⋍14.05 | ||||||
x^2+x+3.25-δ⋍0 x⋍(-0.5)±11[√3-δ]⒤ | |||||||
p⋍(1/2)±11⒤[√(1-s)-(q/11)] | |||||||
文献値 | 文/11 | ( | s | ) | +s/11 | ⋍√2k+1 | ⋍2k+1 |
14.13472 | 1.28497454545455 | ( | 5 | ) | 0.45454545454546 | 1.73952 | 3.026 |
21.02203 | 1.91109363636364 | ( | 4 | ) | 0.36363636363636 | 2.27473 | 5.174 |
25.01085 | 2.27371363636364 | ( | 4 | ) | 0.36363636363636 | 2.63735 | 6.956 |
30.42487 | 2.76589727272727 | ( | 6 | ) | 0.54545454545455 | 3.31135 | 10.965 |
32.93506 | 2.99409636363636 | ( | 7 | ) | 0.63636363636364 | 3.63046 | 13.180 |
37.58617 | 3.41692454545455 | ( | 5 | ) | 0.45454545454546 | 3.87147 | 14.988 |
40.91871 | 3.71988272727273 | ( | 5 | ) | 0.45454545454546 | 4.17443 | 17.426 |
43.32707 | 3.93882454545455 | ( | 5 | ) | 0.45454545454546 | 4.39337 | 19.302 |
48.00515 | 4.36410454545455 | ( | 5 | ) | 0.45454545454546 | 4.81865 | 23.219 |
49.77383 | 4.52489363636364 | ( | 4 | ) | 0.36363636363636 | 4.88853 | 23.898 |
52.97 | 4.81545454545455 | ( | 5 | ) | 0.45454545454546 | 5.27000 | 27.773 |
56.446 | 5.13145454545455 | ( | 5 | ) | 0.45454545454546 | 5.58600 | 31.203 |
59.347 | 5.39518181818182 | ( | 2 | ) | 0.18181818181818 | 5.57700 | 31.103 |
60.831 | 5.53009090909091 | ( | 1 | ) | 0.09090909090909 | 5.62100 | 31.596 |
65.112 | 5.91927272727273 | ( | 3 | ) | 0.27272727272727 | 6.19200 | 38.341 |