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用欧拉恒等式求平方和相等的数对

2014-03-09 20:39阅读：

http://blog.sina.cn/dpool/blog/u/1041457940

用欧拉恒等式求平方和相等的数对
1），欧拉恒等式的原式为
(a²+b²)(c²+d²)=(ac+bd)²+(ad-bc)²=(ac-bd)²+(ad+bc)²
（a>b ,(a,b)=1,a+b≠2k; c>d ,(c,d)=1,c+d≠2k）
即当A=(a²+b²)(c²+d²)两对数平方和乘积时，A可用两对平方和相等的数对表示。
如：(a²+b²)(c²+d²)=(1²+2²)(2²+3²)=65
65= (ac+bd)²+(ad-bc)²=8²+1²
= (ac-bd)²+(ad+bc)²=4²+7²

A=(a²+b²)(c²+d²)
 ac+bd
 ad-bc


ac-bd
 ad+bc


2），增加乘积数对的个数，即是欧拉恒等式的推广形式。当A=(a²+b²)(c²+d²)(e²+f²)三数对平方和乘积时，则A可用四对平方和相等的数对表示。
如：(1²+2²)(1²+4²)(2²+3²)=1105
1105=4²+33²
=9²+32²
=12²+31²
=23²+24²

A=(a²+b²)(c²+d²)(e²+f²)
 (ac+bd)e+(ad-bc)f
 (ac+bd)f-(ad-bc)e

(ac+bd)e-(ad-bc)f
 (ac+bd)f+(ad-bc)e

(ac-bd)e+(ad+bc)f
 (ac-bd)f-(ad+bc)e

(ac-bd)e-(ad+bc)f
 (ac-bd)f+(ad+bc)e


3），当A=(a²+b²)(c²+d²)(e²+f²)(g²+h²)四数对平方和乘积时，则A可用八对平方和相等的数对表示。
如：(1²+2²)(1²+4²)(2²+3²)(2²+5²)=32045
32045=2²+179²
=19²+178²
=46²+173²
=67²+166²
=74²+163²
=86²+157²
=109²+142²
=122²+131²

数对
(ac+bd)(ge+hf)+(ad-bc)(gf-he)
 (ac+bd)(gf-he)-(ad-bc)(ge+hf)
(ac+bd)(ge-hf)+(ad-bc)(gf+he)
 (ac+bd)(gf+he)-(ad-bc)(ge-hf)
(ac+bd)(ge+hf)-(ad-bc)(gf-he)
 (ac+bd)(gf-he)+(ad-bc)(ge+hf)
(ac+bd)(ge-hf)-(ad-bc)(gf+he)
 (ac+bd)(gf+he)+(ad-bc)(ge-hf)
(ac-bd)(ge+hf)+(ad+bc)(gf-he)
 (ac-bd)(gf-he)-(ad+bc)(ge+hf)
(ac-bd)(ge-hf)+(ad+bc)(gf+he)
 (ac-bd)(gf+he)-(ad+bc)(ge-hf)
(ac-bd)(ge+hf)-(ad+bc)(gf-he)
 (ac-bd)(gf-he)+(ad+bc)(ge+hf)
(ac-bd)(ge-hf)-(ad+bc)(gf+he)
 (ac-bd)(gf+he)+(ad+bc)(ge-hf)


4），设A=(a²+b²)(c²+d²)(e²+f²)(g²+h²)(i²+j²)五数对乘积时则可得十六个平方和相等的数对。
如：(1²+2²)(1²+4²)(2²+3²)(2²+5²)(1²+6²)=1185665
1185665=64²+1087²
=103²+1084²
=167²+1076²
=191²+1072²
=236²+1063²
=281+1052²
=292²+1049²
=359²+1028²
=449²+992²
=512²+961²
=568²+929²
=601²+908²
=607²+904²
=664²+863²
=673²+856²
=743²+796²

数对
(ig-jh)[(ac+bd)e+(ad-bc)f]+(ih+jg)[(ac+bd)f-(ad-bc)e]
 (ig-jh)[(ac+bd)f-(ad-bc)e]-(ih+jg)[(ac+bd)e+(ad-bc)f]
(ig+jh)[(ac+bd)e+(ad-bc)f]+(ih-jg)[(ac+bd)f-(ad-bc)e]
 (ig+jh)[(ac+bd)f-(ad-bc)e]-(ih-jg)[(ac+bd)e+(ad-bc)f]
(ig+jh)[(ac+bd)e+(ad-bc)f]-(ih-jh)[(ac+bd)f-(ad-bc)e]
 (ig+jh)[(ac+bd)f-(ad-bc)e]+(ih-jg)[(ac+bd)e+(ad-bc)f]
(ig-jh)[(ac+bd)e+(ad-bc)f]-(ih+jg)[(ac+bd)f-(ad-bc)e]
 (ig-jh)[(ac+bd)f-(ad-bc)e]+(ih+jg)[(ac+bd)e+(ad-bc)f]
(ig-jh)[(ac+bd)e-(ad-bc)f]+(ih+jg)[(ac+bd)f+(ad-bc)e]
 (ig-jh)[(ac+bd)f+(ad-bc)e]-(ih+jg)[(ac+bd)e-(ad-bc)f]
(ig+jh)[(ac+bd)e-(ad-bc)f]+(ih-jg)[(ac+bd)f+(ad-bc)e]
 (ig+jh)[(ac+bd)f+(ad-bc)e]-(ih-jg)[(ac+bd)e-(ad-bc)f]
(ig+jh)[(ac+bd)e-(ad-bc)f]-(ih-jg)[(ac+bd)f+(ad-bc)e]
 (ig+jh)[(ac+bd)f+(ad-bc)e]+(ih-jg)[(ac+bd)e-(ad-bc)f]
(ig-jh)[(ac+bd)e-(ad-bc)f]-(ih+jg)[(ac+bd)f+(ad-bc)e]
 (ig-jh)[(ac+bd)f+(ad-bc)e]+(ih+jg)[(ac+bd)e-(ad-bc)f]
(ig-jh)[(ac-bd)e+(ad+bc)f]+(ih+jg)[(ac-bd)f-(ad+bc)e]
 (ig-jh)[(ac-bd)f-(ad+bc)e]-(ih+jg)[(ac-bd)e+(ad+bc)f]
(ig+jh)[(ac-bd)e+(ad+bc)f]+(ih-jg)[(ac-bd)f-(ad+bc)e]
 (ig+jh)[(ac-bd)f-(ad+bc)e]-(ih-jg)[(ac-bd)e+(ad+bc)f]
(ig+jh)[(ac-bd)e+(ad+bc)f]+(ih+jg)[(ac-bd)f-(ad+bc)e]
 (ig-jh)[(ac-bd)f-(ad+bc)e]+(ih-jg)[(ac-bd)e+(ad+bc)f]
(ig-jh)[(ac-bd)e+(ad+bc)f]+(ih-jg)[(ac-bd)f-(ad+bc)e]
 (ig+jh)[(ac-bd)f-(ad+bc)e]+(ih+jg)[(ac-bd)e+(ad+bc)f]
(ig-jh)[(ac-bd)e-(ad+bc)f]+(ih+jg)[(ac-bd)f+(ad+bc)e]
 (ig-jh)[(ac-bd)f+(ad+bc)e]-(ih+jg)[(ac-bd)e-(ad+bc)f]
(ig+jh)[(ac-bd)e-(ad+bc)f]+(ih-jg)[(ac-bd)f+(ad+bc)e]
 (ig+jh)[(ac-bd)f+(ad+bc)e]-(ih-jg)[(ac-bd)e-(ad+bc)f]
(ig+jh)[(ac-bd)e-(ad+bc)f]-(ih-jg)[(ac-bd)f+(ad+bc)e]
 (ig+jh)[(ac-bd)f+(ad+bc)e]+(ih-jg)[(ac-bd)e-(ad+bc)f]
(ig-jh)[(ac-bd)e-(ad+bc)f]-(ih+jg)[(ac-bd)f+(ad+bc)e]
 (ig-jh)[(ac-bd)f+(ad+bc)e]+(ih+jg)[(ac-bd)e-(ad+bc)f]


当A等于n个数对平方和乘积时，可得2n-1个平方和相等的数对。等号前面部分如有重复数对，则在等号后面也会得到重复数对。在求解幻方时，欧拉恒等式及其推广形式都有很多用途。

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