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楼主: 无心人

[原创] 整点直角三角形

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 楼主| 发表于 2010-6-10 11:46:23 | 显示全部楼层
上面的约束条件太宽,搜索2的出来一堆相同的
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2010-6-10 11:48:10 | 显示全部楼层
结论是
n = 2, 4, 5, 6, 8, 10存在解
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2010-6-10 11:49:22 | 显示全部楼层
11不加额外约束计算就太慢了,估计不存在解
==========================
计算完了,11不存在, 以后只计算奇数的了

另外,合数也可以证明只要其一个质因数存在解,则该合数存在解
=================================
n = 13, 15, 17都存在解
在计算19
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2010-6-10 12:04:19 | 显示全部楼层
14# 无心人
约束:
a^2+b^2+c^2+d^2=n^2
ac+bd=0
(a^2+b^2)(c^2+d^2)!=0
a<=b
c<=d


另外,容易得知:对于非平凡的n,应该形如x^2+y^2
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2010-6-10 12:19:19 | 显示全部楼层
如果n不存在形如 x^2+y^2(x>=1,y>=1)的质因子,那么此题无解

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毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2010-6-10 13:47:37 | 显示全部楼层
形如x^2+y^2的质数有:http://www.research.att.com/~njas/sequences/A002313

所以,无整点解的n,即质因子不含有上述质数,有:
3, 7, 9, 11, 19, 21, 23, 27, 31, 33, 43, 47, 49, 57, 59, 63, 67, 69, 71, 77, 79, 81, 83, 93, 99, 103, 107, 121, 127, 129, 131, 133, 139, 141, 147, 151, 161, 163, 167, 171, 177, 179, 189, 191, 199, 201, 207, 209, 211, 213, 217, 223, 227, 231, 237, 243, 249, 253, 279, 297, 301, 309, 321, 329, 341, 343, 361, 363, 381, 387, 393, 399, 413, 417,
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2010-6-10 14:07:10 | 显示全部楼层
现在来讨论有解的n,设解的个数为s(n),那么:

一、s(2)=4 , s(4)=4,s(8)=4
二、如果质数p不能表示成x^2+y^2的形式,即模4余3,那么 s(n*p)=s(n)

三、 如果p是大于2的质数,能表示成x^2+y^2的形式,即模4余1.那么
s(p)=12
s(p*p)=40
s(2*p)=36
s(4*p)=36
s(2*p*p)=100
s(4*p*p)=100
s(p1*p2)=144 , (p1≠p2)
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
发表于 2010-6-10 14:32:00 | 显示全部楼层
s(125)=84,所有解:

有一个点在原点,另外两个是:

{{{-121, -22}, {-4, 22}}, {{-121, 22}, {4, 22}}, {{-120, 0}, {0, 35}}, {{-117, 0}, {0, 44}}, {{-110, -55}, {-10, 20}}, {{-110, -20}, {-10, 55}}, {{-110, 20}, {10, 55}}, {{-110, 55}, {10, 20}}, {{-100, -50}, {-25, 50}}, {{-100, 0}, {0, 75}}, {{-100, 50}, {25, 50}}, {{-96, -72}, {-21, 28}}, {{-96, -28}, {-21, 72}}, {{-96, 28}, {21, 72}}, {{-96, 72}, {21, 28}}, {{-82, -76}, {-38, 41}}, {{-82, -41}, {-38, 76}}, {{-82, 41}, {38, 76}}, {{-82, 76}, {38, 41}}, {{-80, -60}, {-45, 60}}, {{-80, 60}, {45, 60}}, {{-76, -38}, {-41, 82}}, {{-76, 38}, {41, 82}}, {{-76,  82}, {-41, -38}}, {{-75, 0}, {0, 100}}, {{-72, -21}, {-28,  96}}, {{-72, 21}, {28, 96}}, {{-72, 96}, {-28, -21}}, {{-60, -45}, {-60, 80}}, {{-60, 45}, {60, 80}}, {{-60, 80}, {-60, -45}}, {{-55, -10}, {-20, 110}}, {{-55, 10}, {20, 110}}, {{-55, 110}, {-20, -10}}, {{-50, -25}, {-50, 100}}, {{-50, 25}, {50, 100}}, {{-50, 100}, {-50, -25}}, {{-45, 60}, {-80, -60}}, {{-44, 0}, {0, 117}}, {{-41, -38}, {-76, 82}}, {{-41, 38}, {76, 82}}, {{-41, 82}, {-76, -38}}, {{-38, 41}, {-82, -76}}, {{-38, 76}, {-82, -41}}, {{-35, 0}, {0, 120}}, {{-28, -21}, {-72, 96}}, {{-28, 21}, {72, 96}}, {{-28, 96}, {-72, -21}}, {{-25, 50}, {-100, -50}}, {{-22, -4}, {-22, 121}}, {{-22, 4}, {22, 121}}, {{-22, 121}, {-22, -4}}, {{-21, 28}, {-96, -72}}, {{-21, 72}, {-96, -28}}, {{-20, -10}, {-55, 110}}, {{-20, 10}, {55, 110}}, {{-20, 110}, {-55, -10}}, {{-10, 20}, {-110, -55}}, {{-10, 55}, {-110, -20}}, {{-4, 22}, {-121, -22}}, {{0, 35}, {-120, 0}}, {{0, 44}, {-117, 0}}, {{0, 75}, {-100, 0}}, {{0, 100}, {-75, 0}}, {{0, 117}, {-44, 0}}, {{0, 120}, {-35, 0}}, {{4, 22}, {-121, 22}}, {{10, 20}, {-110, 55}}, {{10, 55}, {-110, 20}}, {{20, 110}, {-55, 10}}, {{21, 28}, {-96, 72}}, {{21, 72}, {-96, 28}}, {{22, 121}, {-22, 4}}, {{25, 50}, {-100, 50}}, {{28, 96}, {-72, 21}}, {{38, 41}, {-82, 76}}, {{38, 76}, {-82, 41}}, {{41, 82}, {-76, 38}}, {{45, 60}, {-80, 60}},{{50, 100}, {-50, 25}}, {{55,  110}, {-20, 10}}, {{60, 80}, {-60, 45}}, {{72, 96}, {-28, 21}}, {{76, 82}, {-41, 38}}}
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2010-6-10 14:48:11 | 显示全部楼层
19确实无解

需要检验的50内奇数有下列数字
21, 23, 27, 31, 33, 37, 41, 43, 47, 49
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
 楼主| 发表于 2010-6-10 14:48:45 | 显示全部楼层
你那些约束我用Haskell很难写出来
毋因群疑而阻独见  毋任己意而废人言
毋私小惠而伤大体  毋借公论以快私情
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