整点直角三角形

无心人

上面的约束条件太宽，搜索2的出来一堆相同的

无心人

结论是
n = 2, 4, 5, 6, 8, 10存在解

无心人

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

wayne

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

wayne

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

wayne

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,

wayne

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)

wayne

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}}}

无心人

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

无心人

你那些约束我用Haskell很难写出来