| 【並列回路の抵抗値】 | ||||
| 1/R0= | 1/R1+1/R2+1/R3= | (R2･R3+R3･R1+R1･R2)/R1･R2･R3 | ||
| ∵ | R2･R3+R3･R1+R1･R2 | |||
| = | (b-c)(c-a)+(a-b)(c-a)+(a-b)(b-c) | |||
| = | bc-c＾2-ab+ac+ac-bc-a^2+ab+ab-b^2-ca+bc | |||
| =- | (a^2+b^2+c^2-ab-bc-ca) | |||
| =-0.5* | (a-b)^2+(b-c)^2+(c-a)^2 | |||
| ﾌｧｲﾝﾏﾝ･ﾊﾟﾗﾒｰﾀ(A-B)(B-C)(C-A)^2/(ABC)＾2= | ||||
| ･[1/k-1/(k+1)][1/ℓ-1/(ℓ+1)][1/m-1/(m+1)] | ||||
| ∵1/AB=1/(A-B)^2･[1/k-1/(k+1)] | ||||
| ①(A-B)/AB=１/(A-B)･[1/k-1/(k+1)] | ||||
| ②(B-C)/BC=１/(B-C)･[1/ℓ-1/(ℓ+1)] | ||||
|
③(C-A)/CA=１/(C-A)･[1/m-1/(m+1)] ∵①ｘ②ｘ③ |
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| Dyson(1968) | ∵①~⑩=288τ(n) | |||
| ∴(①~⑩)/(abcde)^4=(１/①~⑩)・ | ||||
| ･[1/k-1/(k+1)][1/ℓ-1/(ℓ+1)][1/m-1/(m+1)] | ||||
| ∴(①~⑩)＾2/(abcde)^4=[288τ(n)]^2/(abcde)^4 | ||||
| [1/k-1/(k+1)][1/ℓ-1/(ℓ+1)][1/m-1/(m+1)] | ||||
| ∵ | ca(c-a)+ab(a-b)+bc(b-c) | |||
| = | (b-c)a^2+(c^2-b^2)a+bc(b-c) | |||
| = | (b-c)｛a^2-(b+c)+bc｝ | |||
| = | (b-c)(a-b)(a-c) | |||
| = | -(a-b)(b-c)(c-a) | |||
| 両辺をabcで割 | ||||
| (c-a)/b+(a-b)/c+(b-c)/a | ||||
| =- | (a-b)(b-c)(c-a)/abc | |||
| =- | (abc)/(a-b)(b-c)(c-a)･ | |||
| ･[1/k-1/(k+1)][1/ℓ-1/(ℓ+1)][1/m-1/(m+1)] | ||||
| ※ | 1/AB=1/(A-B)^2･[1/k-1/(k+1)] | |||
| ①(A-B)/AB=１/(A-B)･[1/k-1/(k+1)] | ||||
| ②(B-C)/BC=１/(B-C)･[1/ℓ-1/(ℓ+1)] | ||||
| ③(C-A)/CA=１/(C-A)･[1/m-1/(m+1)] | ||||
| ∴(A-B)(B-C)(C-A)/(ABC)＾2=１/(A-B)(B-C)(C-A)･ | ||||
| ･[1/k-1/(k+1)][1/ℓ-1/(ℓ+1)][1/m-1/(m+1)] | ||||
| ∴(A-B)(B-C)(C-A)/(ABC)=(ABC)/(A-B)(B-C)(C-A)･ | ||||
| ･[1/k-1/(k+1)][1/ℓ-1/(ℓ+1)][1/m-1/(m+1)] | ||||
| 1/AB=1/(A-B)^2･[1/k-1/(k+1)] | |||
| ①(A-B)/AB=１/(A-B)･[1/k-1/(k+1)] | |||
| ②(B-C)/BC=１/(B-C)･[1/ℓ-1/(ℓ+1)] | |||
| ③(C-A)/CA=１/(C-A)･[1/m-1/(m+1)] | |||
| ∴(A-B)(B-C)(C-A)/(ABC)＾2=１/(A-B)(B-C)(C-A)･ | |||
| ･[1/k-1/(k+1)][1/ℓ-1/(ℓ+1)][1/m-1/(m+1)] | |||
| Dyson(1968) | ∵1/①~⑩=1/1!2!3!4!・τ(n) | ||
| ∴(①~⑩)/(abcde)^4=(１/①~⑩)・ | |||
| ･[1/k-1/(k+1)][1/ℓ-1/(ℓ+1)][1/m-1/(m+1)] | |||
| n | τ(n) | 素因数 | |
| 1 | 1 | ||
| 2 | - | 24 | 2・2・2・3 |
| 3 | 252 | 2・2・3・3・7 | |
| 4 | - | 1472 | 2・2・2・2・2・2・23 |
| 5 | 4830 | 2・3・5・7・23 | |
| 6 | - | 6048 | 2・2・2・2・2・3・3・3・7 |
| 7 | - | 16744 | 2・2・2・7・13・23 |
| 8 | 84480 | 2＾9・3・5・11 | |
| 9 | - | 113643 | 3・3・3・3・23・61 |
| 10 | - | 115920 | 2・2・2・2・3・3・5・7・23 |
| 1/AB=1/(A-B)^2･[1/k-1/(k+1)] | ||||
| ①(A-B)/AB=１/(A-B)･[1/k-1/(k+1)] | ||||
| ②(B-C)/BC=１/(B-C)･[1/ℓ-1/(ℓ+1)] | ||||
| ③(C-A)/CA=１/(C-A)･[1/m-1/(m+1)] | ||||
| ∴(A-B)(B-C)(C-A)/(ABC)＾2=１/(A-B)(B-C)(C-A)･ | ||||
| ･[1/k-1/(k+1)][1/ℓ-1/(ℓ+1)][1/m-1/(m+1)] | ||||
| n | τ(n) | 素因数 | ||
| 1 | 1 | 1 | ||
| 2 | - | 24 | = | 2・2・2・3 |
| 3 | 252 | = | 2・2・3・3・7 | |
| 4 | - | 1472 | = | 2・2・2・2・2・2・23 |
| 5 | 4830 | = | 2・3・5・7・23 | |
| 6 | - | 6048 | = | 2・2・2・2・2・3・3・3・7 |
| 7 | - | 16744 | = | 2・2・2・7・13・23 |
| 8 | 84480 | = | 2＾9・3・5・11 | |
| 9 | - | 113643 | = | 3・3・3・3・23・61 |
| 10 | - | 115920 | = | 2・2・2・2・3・3・5・7・23 |