『**指數a,b,c,d無上下限:X+Y+Z=1的恆等式**』
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『示意圖』
※※※
[(4+a)(2+b)-(3+b)(3+a)]
×
[(7+c)(3+d)-(6+d)(4+c)]
+
[(5+c)(3+d)-(4+d)(4+c)]
×
[(5+a)(2+b)-(4+b)(3+a)]
=
[(4+a)(2+b)-(3+b)(3+a)]
×
[(6+c)(3+d)-(5+d)(4+c)]
+
[(5+c)(3+d)-(4+d)(4+c)]
×
[(6+a)(2+b)-(5+b)(3+a)]
※※※
解釋：(4+a)=(X^(4+a)+Y^(4+a)+Z^(4+a))，如此類推。
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『指數a,b,c,d無上下限:X+Y+Z=1的恆等式』
(為方便觀看，下式展示a=b=c=d=0的狀態)
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[(X^4+Y^4+Z^4)(X^2+Y^2+Z^2)-(X^3+Y^3+Z^3)(X^3+Y^3+Z^3)]
×
[(X^7+Y^7+Z^7)(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^3+Y^3+Z^3)-(X^4+Y^4+Z^4)(X^4+Y^4+Z^4)]
×
[(X^5+Y^5+Z^5)(X^2+Y^2+Z^2)-(X^4+Y^4+Z^4)(X^3+Y^3+Z^3)]
=
[(X^4+Y^4+Z^4)(X^2+Y^2+Z^2)-(X^3+Y^3+Z^3)(X^3+Y^3+Z^3)]
×
[(X^6+Y^6+Z^6)(X^3+Y^3+Z^3)-(X^5+Y^5+Z^5)(X^4+Y^4+Z^4)]
+
[(X^5+Y^5+Z^5)(X^3+Y^3+Z^3)-(X^4+Y^4+Z^4)(X^4+Y^4+Z^4)]
×
[(X^6+Y^6+Z^6)(X^2+Y^2+Z^2)-(X^5+Y^5+Z^5)(X^3+Y^3+Z^3)]
※※※
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，X+Y+Z=1。
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**範例一：X=2, Y=3, Z= -4，a=b=c=d=0。**
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『範例一的示意圖』
[(4)(2)-(3)(3)]
×
[(7)(3)-(6)(4)]
+
[(5)(3)-(4)(4)]
×
[(5)(2)-(4)(3)]
=
[(4)(2)-(3)(3)]
×
[(6)(3)-(5)(4)]
+
[(5)(3)-(4)(4)]
×
[(6)(2)-(5)(3)]
※※※
解釋：(4)=(X^4+Y^4+Z^4)，如此類推。
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[(2^4+3^4+(-4)^4)(2^2+3^2+(-4)^2)-(2^3+3^3+(-4)^3)(2^3+3^3+(-4)^3)]
×
[(2^7+3^7+(-4)^7)(2^3+3^3+(-4)^3)-(2^6+3^6+(-4)^6)(2^4+3^4+(-4)^4)]
+
[(2^5+3^5+(-4)^5)(2^3+3^3+(-4)^3)-(2^4+3^4+(-4)^4)(2^4+3^4+(-4)^4)]
×
[(2^5+3^5+(-4)^5)(2^2+3^2+(-4)^2)-(2^4+3^4+(-4)^4)(2^3+3^3+(-4)^3)]
=
[(2^4+3^4+(-4)^4)(2^2+3^2+(-4)^2)-(2^3+3^3+(-4)^3)(2^3+3^3+(-4)^3)]
×
[(2^6+3^6+(-4)^6)(2^3+3^3+(-4)^3)-(2^5+3^5+(-4)^5)(2^4+3^4+(-4)^4)]
+
[(2^5+3^5+(-4)^5)(2^3+3^3+(-4)^3)-(2^4+3^4+(-4)^4)(2^4+3^4+(-4)^4)]
×
[(2^6+3^6+(-4)^6)(2^2+3^2+(-4)^2)-(2^5+3^5+(-4)^5)(2^3+3^3+(-4)^3)]
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9,396×(−1,317,816)
+
(−102,888)×(−11,484)
=
9,396×122,616
+
(−102,888)×120,060
※※※
−11,200,633,344
=
−11,200,633,344
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**範例二：X=2, Y=3, Z= -4，a=1, b=3, c=2, d=4。**
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『範例二的示意圖』
[(5)(5)-(6)(4)]
×
[(9)(7)-(10)(6)]
+
[(7)(7)-(8)(6)]
×
[(6)(5)-(7)(4)]
=
[(5)(5)-(6)(4)]
×
[(8)(7)-(9)(6)]
+
[(7)(7)-(8)(6)]
×
[(7)(5)-(8)(4)]
※※※
解釋：(5)=(X^5+Y^5+Z^5)，如此類推。
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[(2^5+3^5+(-4)^5)(2^5+3^5+(-4)^5)-(2^6+3^6+(-4)^6)(2^4+3^4+(-4)^4)]
×
[(2^9+3^9+(-4)^9)(2^7+3^7+(-4)^7)-(2^10+3^10+(-4)^10)(2^6+3^6+(-4)^6)]
+
[(2^7+3^7+(-4)^7)(2^7+3^7+(-4)^7)-(2^8+3^8+(-4)^8)(2^6+3^6+(-4)^6)]
×
[(2^6+3^6+(-4)^6)(2^5+3^5+(-4)^5)-(2^7+3^7+(-4)^7)(2^4+3^4+(-4)^4)]
=
[(2^5+3^5+(-4)^5)(2^5+3^5+(-4)^5)-(2^6+3^6+(-4)^6)(2^4+3^4+(-4)^4)]
×
[(2^8+3^8+(-4)^8)(2^7+3^7+(-4)^7)-(2^9+3^9+(-4)^9)(2^6+3^6+(-4)^6)]
+
[(2^7+3^7+(-4)^7)(2^7+3^7+(-4)^7)-(2^8+3^8+(-4)^8)(2^6+3^6+(-4)^6)]
×
[(2^7+3^7+(-4)^7)(2^5+3^5+(-4)^5)-(2^8+3^8+(-4)^8)(2^4+3^4+(-4)^4)]
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(−1,164,816)×(−2,016,204,480)
+
(−155,797,056)×1,304,496
=
(−1,164,816)×164,954,304
+
(−155,797,056)×(−15,002,928)
※※※
2145270601211904.
=
2145270601211904.
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**範例三：X=3, Y=4, Z= -6，a=1, b=3, c=2, d=4。**
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『範例三的示意圖』
[(5)(5)-(6)(4)]
×
[(9)(7)-(10)(6)]
+
[(7)(7)-(8)(6)]
×
[(6)(5)-(7)(4)]
=
[(5)(5)-(6)(4)]
×
[(8)(7)-(9)(6)]
+
[(7)(7)-(8)(6)]
×
[(7)(5)-(8)(4)]
※※※
解釋：(5)=(X^5+Y^5+Z^5)，如此類推。
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[(3^5+4^5+(-6)^5)(3^5+4^5+(-6)^5)-(3^6+4^6+(-6)^6)(3^4+4^4+(-6)^4)]
×
[(3^9+4^9+(-6)^9)(3^7+4^7+(-6)^7)-(3^10+4^10+(-6)^10)(3^6+4^6+(-6)^6)]
+
[(3^7+4^7+(-6)^7)(3^7+4^7+(-6)^7)-(3^8+4^8+(-6)^8)(3^6+4^6+(-6)^6)]
×
[(3^6+4^6+(-6)^6)(3^5+4^5+(-6)^5)-(3^7+4^7+(-6)^7)(3^4+4^4+(-6)^4)]
=
[(3^5+4^5+(-6)^5)(3^5+4^5+(-6)^5)-(3^6+4^6+(-6)^6)(3^4+4^4+(-6)^4)]
×
[(3^8+4^8+(-6)^8)(3^7+4^7+(-6)^7)-(3^9+4^9+(-6)^9)(3^6+4^6+(-6)^6)]
+
[(3^7+4^7+(-6)^7)(3^7+4^7+(-6)^7)-(3^8+4^8+(-6)^8)(3^6+4^6+(-6)^6)]
×
[(3^7+4^7+(-6)^7)(3^5+4^5+(-6)^5)-(3^8+4^8+(-6)^8)(3^4+4^4+(-6)^4)]
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(−41,701,392)×(−609,583,548,096)
+
(−21,868,273,728)×91,719,216
=
(−41,701,392)×46,464,663,744
+
(−21,868,273,728)×(−1,159,322,544)
※※※
23414741574296592384.
=
23414741574296592384.
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完

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『指數a,b,c,d無上下限:X+Y+Z=1的恆等式』

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