呵呵，有道理，
我再想想，改改程序
本帖最后由 KeyTo9_Fans 于 2010-1-5 16:02 编辑
看看1楼的式子用Tex排出来效果如何:lol
第一个式子：
$\frac{\sqrt{576+6\sqrt{105-6\root{3}{620+12\sqrt{849}}-6\root{3}{620-12\sqrt{849}}}-6\sqrt{8\root{3}{63332+1860\sqrt{849}}+8\root{3}{63332-1860\sqrt{849}}+70\root{3}{620+12\sqrt{849}}+70\root{3}{620-12\sqrt{849}}+969}+6\sqrt{1938-8\root{3}{63332+1860\sqrt{849}}-8\root{3}{63332-1860\sqrt{849}}-70\root{3}{620+12\sqrt{849}}-70\root{3}{620-12\sqrt{849}}+90\sqrt{105-6\root{3}{620+12\sqrt{849}}-6\root{3}{620-12\sqrt{849}}}}-6\sqrt{210+6\root{3}{620+12\sqrt{849}}+6\root{3}{620-12\sqrt{849}}+6\sqrt{8\root{3}{63332+1860\sqrt{849}}+8\root{3}{63332-1860\sqrt{849}}+70\root{3}{620+12\sqrt{849}}+70\root{3}{620-12\sqrt{849}}+969}}}}{6}$
第二个式子：
$\frac{\sqrt{144-3\sqrt{6\root{3}{37484+12\sqrt{849}}+6\root{3}{37484-12\sqrt{849}}+201}-3\sqrt{402-6\root{3}{37484+12\sqrt{849}}-6\root{3}{37484-12\sqrt{849}}+6\sqrt{8\root{3}{175646564+112452\sqrt{849}}+8\root{3}{175646564-112452\sqrt{849}}-134\root{3}{37484+12\sqrt{849}}-134\root{3}{37484-12\sqrt{849}}+9}}}}{3}$
还有3楼的式子：
$\sqrt{16-\sqrt{2/3\root{3}{37484+12\sqrt{849}}+2/3\root{3}{37484-12\sqrt{849}}+67/3}-\sqrt{134/3-2/3\root{3}{37484+12\sqrt{849}}-2/3\root{3}{37484-12\sqrt{849}}+2\sqrt{2/3\root{3}{-108+12\sqrt{849}}-2/3\root{3}{108+12\sqrt{849}}+1}}}$
另外，此贴一发，Fans的积分就达到4位数了。:victory:
本帖最后由 wayne 于 2010-1-5 13:42 编辑
重新编了程序。还是只寻找正整数解的情况。这次快了好几个数量级，找出x<1000的所有正整数解的情况，1秒钟就能搞定
这是前几个
{x,{a,b,c}}
{40, {{401, 58, 38}}},
{56, {{119, 70, 30}}},
{63, {{663, 87, 55}, {105, 87, 35}}},
{70, {{182, 74, 21}}},
{80, {{802, 116, 76}, {116, 100, 35}}},
{96, {{296, 104, 35}}},
{105, {{273, 175,90}, {175, 119, 40}}},
{112, {{238, 140, 60}, {238, 113, 14}}}
{117, {{533, 195, 120}}},
{119, {{7081, 169, 118}}},
{120, {{1203, 174, 114}}},
{126, {{1326, 174, 110}, {210, 174, 70}}},
{140, {{364, 175, 80}, {364, 148, 42}}},
{144, {{219, 156, 44}}},
{160, {{1604, 232, 152}, {232, 200, 70}}},
{165, {{915, 429,275}, {429, 187, 72}}},
{168, {{357, 210, 90}, {210, 182, 45}}},
{189, {{1989, 261, 165}, {315, 261, 105}}},
{192, {{592, 208, 70}}},
{200, {{2005, 290, 190}}}
我之前的那个随机搜索算法搜了十分钟原来是在下图处转圈，前面这么多小的值都给无视了，:L 。。。
{200, {{2005, 290, 190}}} 等同于 {40, {{401, 58, 38}}}
呵,只要x值算出来是前面结果整数倍的均是重复的
例:{40, {{401, 58, 38}}}={80, {{802, 116, 76}}}={120, {{1203, 174, 114}}}={160, {{1604, 232, 152}={200, {{2005, 290, 190}}}....
当然,编程排除掉这些重复项也比较简单...
本帖最后由 wayne 于 2010-1-5 14:41 编辑
有道理
只保留非平凡解，即a,b,c互质的解，x<1000，总共有74组：
{40, {{401, 58, 38}}}
{56, {{119, 70, 30}}}
{63, {{663, 87, 55}, {105, 87, 35}}}
{70, {{182, 74, 21}}}
{80, {{116, 100, 35}}}
{96, {{296, 104, 35}}}
{105, {{273, 175, 90}, {175, 119, 40}}}
{112, {{238, 113, 14}}}
{117, {{533, 195, 120}}}
{119, {{7081, 169, 118}}}
{140, {{364, 175, 80}}}
{144, {{219, 156, 44}}}
{165, {{915, 429, 275}, {429, 187, 72}}}
{168, {{210, 182, 45}}}
{224, {{1799, 476, 340}}}
{225, {{1695, 255, 112}}}
{252, {{1148, 273, 96}, {420, 273, 80}}}
{260, {{1700, 676, 455}}}
{264, {{814, 275, 70}, {561, 286, 90}}}
{275, {{715, 365, 176}}}
{300, {{500, 375, 144}}}
{315, {{1435, 357, 150}}}
{364, {{1300, 455, 224}}}
{385, {{1375, 407, 120}}}
{400, {{1625, 500, 252}}}
{408, {{442, 425, 70}}}
{448, {{3152, 848, 585}}}
{455, {{7969, 481, 153}, {2137, 679, 406}}}
{460, {{1196, 667, 336}}}
{462, {{2562, 770, 495}}}
{483, {{1725, 667, 360}}}
{495, {{3729, 825, 560}, {825, 583, 210}}}
{496, {{2015, 1054, 630}}}
{520, {{1105, 754, 350}, {650, 533, 90}}}
{525, {{1875, 875, 504}}}
{528, {{1628, 660, 315}, {1122, 822, 385}}}
{552, {{1702, 1173, 630}}}
{555, {{4181, 1443, 1008}}}
{576, {{1224, 776, 351}}}
{585, {{4407, 689, 336}, {689, 663, 168}}}
{595, {{3637, 757, 414}, {2125, 1547, 840}, {1547, 1099, 561}}}
{600, {{870, 845, 306}, {750, 625, 126}}}
{612, {{1887, 1020, 560}}}
{616, {{4334, 1309, 910}, {1166, 770, 315}}}
{630, {{6630, 1306, 975}}}
{637, {{28987, 763, 414}}}
{651, {{6851, 1085, 770}}}
{660, {{4972, 1199, 832}}}
{663, {{1937, 1105, 595}}}
{672, {{4728, 840, 455}, {2353, 1022, 574}}}
{675, {{9125, 1125, 819}}}
{680, {{986, 697, 126}}}
{693, {{3843, 707, 135}}}
{728, {{2422, 1547, 858}, {1547, 1378, 630}}}
{735, {{2625, 959, 495}}}
{741, {{4845, 1235, 819}}}
{759, {{1495, 809, 230}}}
{760, {{3838, 950, 495}, {1976, 1615, 800}}}
{765, {{1989, 901, 378}}}
{775, {{9703, 2015, 1560}}}
{792, {{6558, 2442, 1705}, {2442, 1067, 546}}}
{800, {{4040, 1160, 693}}}
{819, {{15981, 3731, 2964}, {15981, 1869, 1520}, {5355, 981, 490}}}
{825, {{1625, 1375, 616}}}
{880, {{1276, 884, 77}, {1100, 979, 260}}}
{918, {{23418, 2682, 2275}}}
{920, {{4646, 943, 198}}}
{931, {{22819, 3325, 2800}}}
{936, {{2886, 2210, 1155}, {1560, 975, 224}}}
{945, {{21273, 2639, 2208}, {1233, 1071, 308}}}
{952, {{1802, 1073, 374}}}
{975, {{1455, 1105, 351}}}
{980, {{4949, 2548, 1584}}}
{992, {{7967, 1240, 680}}}
wayne 能否把你编写的Mathematica程序贴上来,让我们也学习学习...
3楼说BC的长可以表示成
$\sqrt{16-\sqrt(y)-\sqrt{67-y-sqrt{z}}}$
其中y和z的值确实是那两个三次方程的实根。
可是，$67-y-sqrt{z}<0$啊，再开平方不就成虚数了么？
你可能会说最后结果是实数，可是虚部不为0的数开平方是得不到实数的啊。
是不是你最后一步算错了？
12楼抄了你的式子，发现和1楼的式子对不上。
当年也得到过{56, {119, 70, 30}}。
参见
http://tieba.baidu.com/f?kz=74349506
不过现在才知道原来有那么多整数解。
谢谢wayne了。
cds1 = DeleteCases; tt = (Length - 1)/2;
data = Select}][], EvenQ[#[] - #[]] &];
If >= 2, {ii, Table}, {}], {ii, 2, 500}], {}];cds2 = Table[{ii[], tmp = Sqrt], {2}]^2 - ii[]^2];
Flatten /@ Thread@{Subsets], {2}], Map &, 1/tmp]}}, {ii, cds1}];ans = DeleteCases], Select], And @@ IntegerQ /@ # &]}, {ii, cds2}], {_, {}}];ans1 = DeleteCases], Select], GCD @@ # == 1 &]}, {ii, ans}], {_, {}}]; Table[{"x=", Framed], RoundingRadius -> 15, FrameMargins -> 5, Background -> Green], "\t(a,b,c)=", Sequence @@ Row /@ Map], {-1}]}, {ii, ans1}] // Grid
18# KeyTo9_Fans
原来你提出这题，这蓄谋已久啊~~