『**指數a,b,c,d無上下限:XYZ=1的恆等式**』
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『示意圖』
※※※
[(3+d)(6+b)-(5+b)(4+d)]
×
[(6+c)(6+a)-(7+a)(5+c)]
+
[(4+c)(6+a)-(5+a)(5+c)]
×
[(2+d)(6+b)-(4+b)(4+d)]
=
[(3+d)(6+b)-(5+b)(4+d)]
×
[(3+c)(6+a)-(4+a)(5+c)]
+
[(4+c)(6+a)-(5+a)(5+c)]
×
[(5+d)(6+b)-(7+b)(4+d)]
※※※
解釋：(3+d)=(X^(3+d)+Y^(3+d)+Z^(3+d))，如此類推。
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『指數abcd無上下限:XYZ=1的恆等式』
(為方便觀看，下式展示a=b=c=d=0的狀態)
※※※
[(X^3+Y^3+Z^3)(X^6+Y^6+Z^6)-(X^5+Y^5+Z^5)(X^4+Y^4+Z^4)]
×
[(X^6+Y^6+Z^6)(X^6+Y^6+Z^6)-(X^7+Y^7+Z^7)(X^5+Y^5+Z^5)]
+
[(X^4+Y^4+Z^4)(X^6+Y^6+Z^6)-(X^5+Y^5+Z^5)(X^5+Y^5+Z^5)]
×
[(X^2+Y^2+Z^2)(X^6+Y^6+Z^6)-(X^4+Y^4+Z^4)(X^4+Y^4+Z^4)]
=
[(X^3+Y^3+Z^3)(X^6+Y^6+Z^6)-(X^5+Y^5+Z^5)(X^4+Y^4+Z^4)]
×
[(X^3+Y^3+Z^3)(X^6+Y^6+Z^6)-(X^4+Y^4+Z^4)(X^5+Y^5+Z^5)]
+
[(X^4+Y^4+Z^4)(X^6+Y^6+Z^6)-(X^5+Y^5+Z^5)(X^5+Y^5+Z^5)]
×
[(X^5+Y^5+Z^5)(X^6+Y^6+Z^6)-(X^7+Y^7+Z^7)(X^4+Y^4+Z^4)]
※※※
說明；
(1) 恆等式等號左右：mA+nB=mC+nD。
(2) 成立的條件：a,b,c,d任意，XYZ=1。
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**範例一：X=1/2, Y=2/3, Z=3，a=b=c=d=0。**
※※※
『範例一的示意圖』
[(3)(6)-(5)(4)]
×
[(6)(6)-(7)(5)]
+
[(4)(6)-(5)(5)]
×
[(2)(6)-(4)(4)]
=
[(3)(6)-(5)(4)]
×
[(3)(6)-(4)(5)]
+
[(4)(6)-(5)(5)]
×
[(5)(6)-(7)(4)]
※※※
解釋：(3)=(X^3+Y^3+Z^3)，如此類推。
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(((1/2)^3+(2/3)^3+3^3)((1/2)^6+(2/3)^6+3^6)-((1/2)^5+(2/3)^5+3^5)((1/2)^4+(2/3)^4+3^4))
×
(((1/2)^6+(2/3)^6+3^6)((1/2)^6+(2/3)^6+3^6)-((1/2)^7+(2/3)^7+3^7)((1/2)^5+(2/3)^5+3^5))
+
(((1/2)^4+(2/3)^4+3^4)((1/2)^6+(2/3)^6+3^6)-((1/2)^5+(2/3)^5+3^5)((1/2)^5+(2/3)^5+3^5))
×
(((1/2)^2+(2/3)^2+3^2)((1/2)^6+(2/3)^6+3^6)-((1/2)^4+(2/3)^4+3^4)((1/2)^4+(2/3)^4+3^4))
=
(((1/2)^3+(2/3)^3+3^3)((1/2)^6+(2/3)^6+3^6)-((1/2)^5+(2/3)^5+3^5)((1/2)^4+(2/3)^4+3^4))
×
(((1/2)^3+(2/3)^3+3^3)((1/2)^6+(2/3)^6+3^6)-((1/2)^4+(2/3)^4+3^4)((1/2)^5+(2/3)^5+3^5))
+
(((1/2)^4+(2/3)^4+3^4)((1/2)^6+(2/3)^6+3^6)-((1/2)^5+(2/3)^5+3^5)((1/2)^5+(2/3)^5+3^5))
×
(((1/2)^5+(2/3)^5+3^5)((1/2)^6+(2/3)^6+3^6)-((1/2)^7+(2/3)^7+3^7)((1/2)^4+(2/3)^4+3^4))
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233.53302897805×(−221.6832740340)
+
118.75207904663×465.05994941700
=
233.53302897805×233.53302897805
+
118.75207904663×(−430.1499949988)
※※※
3456.4694156480
=
3456.4694156480
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**範例二：X=1/2, Y=2/3, Z=3, a=0, b=1, c=0, d=2。**
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『範例二的示意圖』
[(5)(7)-(6)(6)]
×
[(6)(6)-(7)(5)]
+
[(4)(6)-(5)(5)]
×
[(4)(7)-(5)(6)]
=
[(5)(7)-(6)(6)]
×
[(3)(6)-(4)(5)]
+
[(4)(6)-(5)(5)]
×
[(7)(7)-(8)(6)]
※※※
解釋：(5)=(X^5+Y^5+Z^5)，如此類推。
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(((1/2)^5+(2/3)^5+3^5)((1/2)^7+(2/3)^7+3^7)-((1/2)^6+(2/3)^6+3^6)((1/2)^6+(2/3)^6+3^6))
×
(((1/2)^6+(2/3)^6+3^6)((1/2)^6+(2/3)^6+3^6)-((1/2)^7+(2/3)^7+3^7)((1/2)^5+(2/3)^5+3^5))
+
(((1/2)^4+(2/3)^4+3^4)((1/2)^6+(2/3)^6+3^6)-((1/2)^5+(2/3)^5+3^5)((1/2)^5+(2/3)^5+3^5))
×
(((1/2)^4+(2/3)^4+3^4)((1/2)^7+(2/3)^7+3^7)-((1/2)^5+(2/3)^5+3^5)((1/2)^6+(2/3)^6+3^6))
=
(((1/2)^5+(2/3)^5+3^5)((1/2)^7+(2/3)^7+3^7)-((1/2)^6+(2/3)^6+3^6)((1/2)^6+(2/3)^6+3^6))
×
(((1/2)^3+(2/3)^3+3^3)((1/2)^6+(2/3)^6+3^6)-((1/2)^4+(2/3)^4+3^4)((1/2)^5+(2/3)^5+3^5))
+
(((1/2)^4+(2/3)^4+3^4)((1/2)^6+(2/3)^6+3^6)-((1/2)^5+(2/3)^5+3^5)((1/2)^5+(2/3)^5+3^5))
×
(((1/2)^7+(2/3)^7+3^7)((1/2)^7+(2/3)^7+3^7)-((1/2)^8+(2/3)^8+3^8)((1/2)^6+(2/3)^6+3^6))
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221.68327403406×(−221.68327403406)
+
118.75207904663×430.14999499885
=
221.68327403406×233.53302897805
+
118.75207904663×(−419.6358887983)
※※※
1937.7322215533
=
1937.7322215533
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**範例三：X=1/2, Y=2/3, Z=3, a=1, b=2, c=3, d=5。**
『範例三示意圖』
※※※
[(8)(8)-(7)(9)]
×
[(9)(7)-(8)(8)]
+
[(7)(7)-(6)(8)]
×
[(7)(8)-(6)(9)]
=
[(8)(8)-(7)(9)]
×
[(6)(7)-(5)(8)]
+
[(7)(7)-(6)(8)]
×
[(10)(8)-(9)(9)]
※※※
解釋：(8)=(X^8+Y^8+Z^8)，如此類推。
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(((1/2)^8+(2/3)^8+3^8)((1/2)^8+(2/3)^8+3^8)-((1/2)^7+(2/3)^7+3^7)((1/2)^9+(2/3)^9+3^9))
×
(((1/2)^9+(2/3)^9+3^9)((1/2)^7+(2/3)^7+3^7)-((1/2)^8+(2/3)^8+3^8)((1/2)^8+(2/3)^8+3^8))
+
(((1/2)^7+(2/3)^7+3^7)((1/2)^7+(2/3)^7+3^7)-((1/2)^6+(2/3)^6+3^6)((1/2)^8+(2/3)^8+3^8))
×
(((1/2)^7+(2/3)^7+3^7)((1/2)^8+(2/3)^8+3^8)-((1/2)^6+(2/3)^6+3^6)((1/2)^9+(2/3)^9+3^9))
=
(((1/2)^8+(2/3)^8+3^8)((1/2)^8+(2/3)^8+3^8)-((1/2)^7+(2/3)^7+3^7)((1/2)^9+(2/3)^9+3^9))
×
(((1/2)^6+(2/3)^6+3^6)((1/2)^7+(2/3)^7+3^7)-((1/2)^5+(2/3)^5+3^5)((1/2)^8+(2/3)^8+3^8))
+
(((1/2)^7+(2/3)^7+3^7)((1/2)^7+(2/3)^7+3^7)-((1/2)^6+(2/3)^6+3^6)((1/2)^8+(2/3)^8+3^8))
×
(((1/2)^10+(2/3)^10+3^10)((1/2)^8+(2/3)^8+3^8)-((1/2)^9+(2/3)^9+3^9)((1/2)^9+(2/3)^9+3^9))
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(−803.6760109652)×803.6760109652
+
(−419.6358887983)×(−1526.799595959)
=
(−803.6760109652)×(−804.9282294286)
+
(−419.6358887983)×1553.9584460740
※※※
−5195.225133564
=
−5195.225133564
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完

『指數a,b,c,d無上下限:XYZ=1的恆等式』

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