**齊次連環恆等式**
◆◆◆◆◆◆
**[**(X^2+Y^2)(X^3+Y^3)-(X^1+Y^1)(X^4+Y^4)**][**(X^1+Y^1)(X^4+Y^4)-(X^2+Y^2)(X^3+Y^3)**][**(X^1+Y^1)(X^5+Y^5)-(X^2+Y^2)(X^4+Y^4)**]**
=
**[**(X^1+Y^1)(X^5+Y^5)-(X^3+Y^3)^2**][**(X^2+Y^2)^2-(X^1+Y^1)(X^3+Y^3)**][**(X^1+Y^1)(X^5+Y^5)-(X^2+Y^2)(X^4+Y^4)**]**
=
**[**(X^1+Y^1)(X^6+Y^6)-(X^3+Y^3)(X^4+Y^4)**][**(X^2+Y^2)^2-(X^1+Y^1)(X^3+Y^3)**][**(X^1+Y^1)(X^4+Y^4)-(X^2+Y^2)(X^3+Y^3)**]**
※※※
★特徵如下；
※※※
(1) 恆等式只含2項：X,Y。
(2) 無約束條件：X,Y任意。
(3) 全部指數可以同步無上限n次遞增，或者同步無下限n次遞減。
◆◆◆**範例**◆◆◆
令X=2,Y=3；
※※※
((2^2+3^2)(2^3+3^3)-(2^1+3^1)(2^4+3^4))((2^1+3^1)(2^4+3^4)-(2^2+3^2)(2^3+3^3))((2^1+3^1)(2^5+3^5)-(2^2+3^2)(2^4+3^4))
=
((2^1+3^1)(2^5+3^5)-(2^3+3^3)^2)((2^2+3^2)^2-(2^1+3^1)(2^3+3^3))((2^1+3^1)(2^5+3^5)-(2^2+3^2)(2^4+3^4))
=
((2^1+3^1)(2^6+3^6)-(2^3+3^3)(2^4+3^4))((2^2+3^2)^2-(2^1+3^1)(2^3+3^3))((2^1+3^1)(2^4+3^4)-(2^2+3^2)(2^3+3^3))
※※※
−102,600
=
−102,600
=
−102,600
◆◆◆
**指數同步+1；**
※※※
((2^3+3^3)(2^4+3^4)-(2^2+3^2)(2^5+3^5))((2^2+3^2)(2^5+3^5)-(2^3+3^3)(2^4+3^4))((2^2+3^2)(2^6+3^6)-(2^3+3^3)(2^5+3^5))
=
((2^2+3^2)(2^6+3^6)-(2^4+3^4)^2)((2^3+3^3)^2-(2^2+3^2)(2^4+3^4))((2^2+3^2)(2^6+3^6)-(2^3+3^3)(2^5+3^5))
=
((2^2+3^2)(2^7+3^7)-(2^4+3^4)(2^5+3^5))((2^3+3^3)^2-(2^2+3^2)(2^4+3^4))((2^2+3^2)(2^5+3^5)-(2^3+3^3)(2^4+3^4))
※※※
−22,161,600
=
−22,161,600
=
−22,161,600
◆◆◆
**上式指數同步-6；**
※※※
((2^-3+3^-3)(2^-2+3^-2)-(2^-4+3^-4)(2^-1+3^-1))((2^-4+3^-4)(2^-1+3^-1)-(2^-3+3^-3)(2^-2+3^-2))((2^-4+3^-4)(2^0+3^0)-(2^-3+3^-3)(2^-1+3^-1))
=
((2^-4+3^-4)(2^0+3^0)-(2^-2+3^-2)^2)((2^-3+3^-3)^2-(2^-4+3^-4)(2^-2+3^-2))((2^-4+3^-4)(2^0+3^0)-(2^-3+3^-3)(2^-1+3^-1))
=
((2^-4+3^-4)(2^1+3^1)-(2^-2+3^-2)(2^-1+3^-1))((2^-3+3^-3)^2-(2^-4+3^-4)(2^-2+3^-2))((2^-4+3^-4)(2^-1+3^-1)-(2^-3+3^-3)(2^-2+3^-2))
※※※
−0.00000021821198754
= 
−0.00000021821198754
=
−0.00000021821198754
◆◆◆
**上式指數不變：X=2,Y= -5。**
※※※
((2^-3+(-5)^-3)(2^-2+(-5)^-2)-(2^-4+(-5)^-4)(2^-1+(-5)^-1))((2^-4+(-5)^-4)(2^-1+(-5)^-1)-(2^-3+(-5)^-3)(2^-2+(-5)^-2))((2^-4+(-5)^-4)(2^0+(-5)^0)-(2^-3+(-5)^-3)(2^-1+(-5)^-1))
=
((2^-4+(-5)^-4)(2^0+(-5)^0)-(2^-2+(-5)^-2)^2)((2^-3+(-5)^-3)^2-(2^-4+(-5)^-4)(2^-2+(-5)^-2))((2^-4+(-5)^-4)(2^0+(-5)^0)-(2^-3+(-5)^-3)(2^-1+(-5)^-1))
=
((2^-4+(-5)^-4)(2^1+(-5)^1)-(2^-2+(-5)^-2)(2^-1+(-5)^-1))((2^-3+(-5)^-3)^2-(2^-4+(-5)^-4)(2^-2+(-5)^-2))((2^-4+(-5)^-4)(2^-1+(-5)^-1)-(2^-3+(-5)^-3)(2^-2+(-5)^-2))
※※※
−0.000020117979
= 
−0.000020117979
=
−0.000020117979
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**上式X,Y不變，指數同步：+8。**
※※※
((2^5+(-5)^5)(2^6+(-5)^6)-(2^4+(-5)^4)(2^7+(-5)^7))((2^4+(-5)^4)(2^7+(-5)^7)-(2^5+(-5)^5)(2^6+(-5)^6))((2^4+(-5)^4)(2^8+(-5)^8)-(2^5+(-5)^5)(2^7+(-5)^7))
=
((2^4+(-5)^4)(2^8+(-5)^8)-(2^6+(-5)^6)^2)((2^5+(-5)^5)^2-(2^4+(-5)^4)(2^6+(-5)^6))((2^4+(-5)^4)(2^8+(-5)^8)-(2^5+(-5)^5)(2^7+(-5)^7))
=
((2^4+(-5)^4)(2^9+(-5)^9)-(2^6+(-5)^6)(2^7+(-5)^7))((2^5+(-5)^5)^2-(2^4+(-5)^4)(2^6+(-5)^6))((2^4+(-5)^4)(2^7+(-5)^7)-(2^5+(-5)^5)(2^6+(-5)^6))
※※※
−2.0117979E19
= 
−2.0117979E19
=
−2.0117979E19
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完成：大年初七。

齊次連環恆等式

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