这样的数组有多少？

northwolves · 发表于 2024-3-24 11:08:19

楼主证明以下几个即可：{{3, 39, 75, 81}, {3, 63, 69, 75}, {6, 42, 66, 78}, {9, 15, 51, 87}, {9, 33, 69, 75}, {12, 24, 48, 84}, {15, 21, 27, 87}, {15, 21, 57, 81}, {15, 51, 57, 63}, {27, 33, 39, 75}}

mathe · 发表于 2024-3-25 10:24:19

mathe 发表于 2024-3-24 11:02
对于权重为8的本源解，有两类，分别为
i) R7:R3, 也就是8个单位根角度为$x+\omega_7,x+2\omega_7,x+3\omega ...

9#给出了解还有些包含某个数据为0，或者某两个数相等等情况，或者交换某些数后两者等价的存在。
为此，程序进一步对上面数据进行过滤，得到如下附件结果

sv.tgz (2.66 KB, 下载次数: 0)

其中不含参数的常数部分在文件sv.e?u,比如
下面把这些不含参数解全部列出，和前面找出28个特殊解中匹配的全部用{}标出，并且加上前面部分用0开始的索引值
sv.e1u有32个解:
2a=1/12,2b=17/84,2c=43/84,2d=79/84
2a=1/12,2b=19/84,2c=29/84,2d=31/84
2a=1/12,2b=19/84,2c=53/84,2d=55/84
2a=1/12,2b=29/84,2c=53/84,2d=65/84
2a=1/12,2b=31/84,2c=55/84,2d=65/84
2a=1/12,2b=41/84,2c=67/84,2d=79/84
2a=11/84,2b=13/84,2c=23/84,2d=5/12
2a=11/84,2b=13/84,2c=7/12,2d=61/84
2a=11/84,2b=23/84,2c=7/12,2d=71/84
2a=11/84,2b=5/12,2c=61/84,2d=71/84
2a=13/84,2b=23/84,2c=7/12,2d=73/84
2a=13/84,2b=5/12,2c=61/84,2d=73/84
2a=17/84,2b=41/84,2c=11/12,2d=79/84
2a=1/84,2b=25/84,2c=47/84,2d=7/12
2a=1/84,2b=25/84,2c=5/12,2d=37/84
2a=1/84,2b=37/84,2c=7/12,2d=59/84
2a=1/84,2b=5/12,2c=47/84,2d=59/84
2a=19/84,2b=29/84,2c=53/84,2d=11/12
2a=19/84,2b=31/84,2c=55/84,2d=11/12
2a=23/84,2b=5/12,2c=71/84,2d=73/84
2a=25/84,2b=37/84,2c=7/12,2d=83/84
2a=25/84,2b=5/12,2c=47/84,2d=83/84
2a=29/84,2b=31/84,2c=65/84,2d=11/12
2a=43/84,2b=67/84,2c=11/12,2d=79/84
2a=47/84,2b=7/12,2c=59/84,2d=83/84
2a=5/12,2b=37/84,2c=59/84,2d=83/84
2a=53/84,2b=55/84,2c=65/84,2d=11/12
2a=5/84,2b=1/12,2c=17/84,2d=41/84
2a=5/84,2b=1/12,2c=43/84,2d=67/84
2a=5/84,2b=17/84,2c=43/84,2d=11/12
2a=5/84,2b=41/84,2c=67/84,2d=11/12
2a=7/12,2b=61/84,2c=71/84,2d=73/84
sv.e2u也有32个解:
2a=11/20,2b=41/60,2c=53/60,2d=11/12
2a=11/20,2b=7/12,2c=47/60,2d=59/60
2a=1/12,2b=17/60,2c=31/60,2d=19/20
2a=1/12,2b=19/60,2c=11/20,2d=53/60
2a=1/12,2b=29/60,2c=43/60,2d=19/20
2a=1/12,2b=7/60,2c=11/20,2d=41/60
2a=1/12,2b=7/60,2c=19/60,2d=9/20 {25}
2a=1/12,2b=9/20,2c=41/60,2d=53/60
2a=11/60,2b=23/60,2c=7/12,2d=19/20
2a=11/60,2b=5/12,2c=37/60,2d=19/20
2a=1/20,2b=1/12,2c=17/60,2d=29/60 {13}
2a=1/20,2b=1/12,2c=31/60,2d=43/60
2a=1/20,2b=11/60,2c=23/60,2d=5/12 {15}
2a=1/20,2b=11/60,2c=7/12,2d=37/60
2a=1/20,2b=17/60,2c=31/60,2d=11/12
2a=1/20,2b=23/60,2c=7/12,2d=49/60
2a=1/20,2b=29/60,2c=43/60,2d=11/12
2a=1/20,2b=5/12,2c=37/60,2d=49/60
2a=13/60,2b=5/12,2c=11/20,2d=59/60
2a=13/60,2b=9/20,2c=7/12,2d=59/60
2a=1/60,2b=13/60,2c=11/20,2d=7/12
2a=1/60,2b=13/60,2c=5/12,2d=9/20 {3}
2a=1/60,2b=5/12,2c=11/20,2d=47/60
2a=1/60,2b=9/20,2c=7/12,2d=47/60
2a=17/60,2b=29/60,2c=11/12,2d=19/20
2a=19/60,2b=9/20,2c=53/60,2d=11/12
2a=23/60,2b=5/12,2c=49/60,2d=19/20
2a=31/60,2b=43/60,2c=11/12,2d=19/20
2a=5/12,2b=9/20,2c=47/60,2d=59/60
2a=7/12,2b=37/60,2c=49/60,2d=19/20
2a=7/60,2b=19/60,2c=11/20,2d=11/12
2a=7/60,2b=9/20,2c=41/60,2d=11/12
sv.e3u也32个解
2a=1/12,2b=17/60,2c=13/20,2d=41/60
2a=1/12,2b=17/60,2c=19/60,2d=7/20 {27}
2a=1/12,2b=19/60,2c=13/20,2d=43/60
2a=1/12,2b=29/60,2c=17/20,2d=53/60
2a=1/12,2b=3/20,2c=31/60,2d=53/60
2a=1/12,2b=7/20,2c=41/60,2d=43/60
2a=1/12,2b=7/60,2c=31/60,2d=17/20
2a=1/12,2b=7/60,2c=3/20,2d=29/60 {24}
2a=11/60,2b=13/60,2c=7/12,2d=17/20
2a=11/60,2b=5/12,2c=47/60,2d=17/20
2a=13/20,2b=41/60,2c=43/60,2d=11/12
2a=13/60,2b=5/12,2c=49/60,2d=17/20
2a=1/60,2b=23/60,2c=7/12,2d=13/20
2a=1/60,2b=5/12,2c=37/60,2d=13/20
2a=1/60,2b=7/20,2c=23/60,2d=5/12 {5}
2a=1/60,2b=7/20,2c=7/12,2d=37/60
2a=17/60,2b=19/60,2c=13/20,2d=11/12
2a=17/60,2b=7/20,2c=41/60,2d=11/12
2a=19/60,2b=7/20,2c=43/60,2d=11/12
2a=23/60,2b=5/12,2c=13/20,2d=59/60
2a=31/60,2b=17/20,2c=53/60,2d=11/12
2a=3/20,2b=11/60,2c=13/60,2d=5/12 {37}
2a=3/20,2b=11/60,2c=7/12,2d=47/60
2a=3/20,2b=13/60,2c=7/12,2d=49/60
2a=3/20,2b=29/60,2c=53/60,2d=11/12
2a=3/20,2b=5/12,2c=47/60,2d=49/60
2a=7/12,2b=37/60,2c=13/20,2d=59/60
2a=7/12,2b=47/60,2c=49/60,2d=17/20
2a=7/20,2b=23/60,2c=7/12,2d=59/60
2a=7/20,2b=5/12,2c=37/60,2d=59/60
2a=7/60,2b=29/60,2c=17/20,2d=11/12
2a=7/60,2b=3/20,2c=31/60,2d=11/12
sv.e4u也32个解
2a=11/120,2b=17/120,2c=61/120,2d=113/120
2a=11/120,2b=59/120,2c=103/120,2d=113/120
2a=1/120,2b=43/120,2c=49/120,2d=53/120
2a=1/120,2b=43/120,2c=67/120,2d=71/120
2a=1/120,2b=49/120,2c=67/120,2d=77/120
2a=1/120,2b=53/120,2c=71/120,2d=77/120
2a=13/120,2b=31/120,2c=37/120,2d=41/120
2a=13/120,2b=31/120,2c=79/120,2d=83/120
2a=13/120,2b=37/120,2c=79/120,2d=89/120
2a=13/120,2b=41/120,2c=83/120,2d=89/120
2a=17/120,2b=59/120,2c=109/120,2d=113/120
2a=19/120,2b=23/120,2c=29/120,2d=47/120
2a=19/120,2b=23/120,2c=73/120,2d=91/120
2a=19/120,2b=29/120,2c=73/120,2d=97/120
2a=19/120,2b=47/120,2c=91/120,2d=97/120
2a=23/120,2b=29/120,2c=73/120,2d=101/120
2a=23/120,2b=47/120,2c=91/120,2d=101/120
2a=29/120,2b=47/120,2c=97/120,2d=101/120
2a=31/120,2b=37/120,2c=79/120,2d=107/120
2a=31/120,2b=41/120,2c=83/120,2d=107/120
2a=37/120,2b=41/120,2c=89/120,2d=107/120
2a=43/120,2b=49/120,2c=67/120,2d=119/120
2a=43/120,2b=53/120,2c=71/120,2d=119/120
2a=49/120,2b=53/120,2c=77/120,2d=119/120
2a=61/120,2b=103/120,2c=109/120,2d=113/120
2a=67/120,2b=71/120,2c=77/120,2d=119/120
2a=7/120,2b=11/120,2c=17/120,2d=59/120
2a=7/120,2b=11/120,2c=61/120,2d=103/120
2a=7/120,2b=17/120,2c=61/120,2d=109/120
2a=7/120,2b=59/120,2c=103/120,2d=109/120
2a=73/120,2b=91/120,2c=97/120,2d=101/120
2a=79/120,2b=83/120,2c=89/120,2d=107/120
sv.e5u有16个解
2a=11/30,2b=13/30,2c=23/30,2d=29/30
2a=1/15,2b=2/15,2c=4/15,2d=7/15 {19}
2a=1/15,2b=2/15,2c=8/15,2d=11/15
2a=1/15,2b=4/15,2c=8/15,2d=13/15
2a=1/15,2b=7/15,2c=11/15,2d=13/15
2a=1/30,2b=11/30,2c=17/30,2d=23/30
2a=1/30,2b=13/30,2c=19/30,2d=23/30
2a=1/30,2b=7/30,2c=11/30,2d=13/30 {9}
2a=1/30,2b=7/30,2c=17/30,2d=19/30
2a=17/30,2b=19/30,2c=23/30,2d=29/30
2a=2/15,2b=4/15,2c=8/15,2d=14/15
2a=2/15,2b=7/15,2c=11/15,2d=14/15
2a=4/15,2b=7/15,2c=13/15,2d=14/15
2a=7/30,2b=11/30,2c=17/30,2d=29/30
2a=7/30,2b=13/30,2c=19/30,2d=29/30
2a=8/15,2b=11/15,2c=13/15,2d=14/15
而sv.e6u（模式6常数解)只余下6个解
2a=1/12,2b=1/4,2c=7/12,2d=3/4
2a=1/24,2b=3/8,2c=5/8,2d=17/24
2a=1/4,2b=5/12,2c=3/4,2d=11/12
2a=1/8,2b=11/24,2c=19/24,2d=7/8
2a=1/8,2b=5/24,2c=13/24,2d=7/8
2a=7/24,2b=3/8,2c=5/8,2d=23/24

上面这些应该就是所有不含参数的解了。
余下含参数的解有sv.o6 (模式6参数解)
Solution candiate with MOD 6:
2a = -1V2+2V3+3
2b = -1V2+2V3+5
2c = -1V2+2V3+1
2d = +1V3+0
Constrain:
+1V1+1V2-2V3+3 Mod 6=0
+3V2-5V3=0
根据最后一条约束，可以假设
V2=5t
V3=3t，然后代入第一条约束得出
V1-t+3Mod6=0
V1=t-3Mod6
代回前面条件，有
2a=t+3Mod6
2b=5+5Mod6
2c=t+1Mod6
2d=3t
即得出$ \tan(t+\frac{3\pi}{6})\tan(t+\frac{5\pi}{6})\tan(t+\frac{\pi}{6})\tan(3t)=1$
这个正好对应tan(3x)公式导出的解。

不过模式7的含参数解还有202种，可能有些之间还是等价的。前面几组是
Solution candiate with MOD 2:
2a = +1V2-1V3+1V4+0
2b = +1V2-1V3+1V4+1
2c = +1V2+0
2d = +1V2+1
Constrain:
+1V1-1V2+1V3-1V4=0
+3V2-2V3+1V4=0
Solution candiate with MOD 2:
2a = +1V4+1
2b = +1V4+0
2c = -1V4+0
2d = -1V4+1
Constrain:
+1V1-1V4+1 Mod 2=0
+1V2-1V3+1 Mod 2=0
+1V3+1V4+1 Mod 2=0
Solution candiate with MOD 2:
2a = +1V2+1V3-1V4+1
2b = +1V2+1V3-1V4+0
2c = +1V2+0
2d = +1V3+0
Constrain:
+1V1-1V2-1V3+1V4+1 Mod 2=0
+2V2+2V3-2V4=0

其中第一组和第二组都是a,b互为余角，c,d互为余角。当然这两个解的确不等价，它们约束不同，第二组可以看成第一组一种特殊情况。
第三组要求V2+V3-V4是平角或周角，对应2a,2b是平角或周角，也就是a或b一个为平角一个为直角，属于增根。
通过简单代码处理可以将4个角两两互为余角的过滤，然后发现还余下64组，全部是可以找出一个参数组合的两倍为0，而结果要求一对角分别是这个参数组合和它加平角，对应上面增根情况。

所以也就是除掉两对角互为余角的情况和三倍角公式得出的情况，所有特殊情况的有理解只有sv.e?u列出的情况。

王守恩 · 发表于 2024-3-25 17:02:59

这题目要搞个无限模式是不可以的, 我们就来搞个尽可能长的有限模式试试。

在1--89这89个整数度数中选尽可能多的数(每个数最多选1次), 约定任意2个数的和≠90。

满足 $\tan(a_{1})*\tan(a_{2})*\tan(a_{3})*\tan(a_{4})*\cdots*\tan(a_{n})=1$, n 最大是多少?

mathe · 发表于 2024-3-26 06:37:49

应该对于任意奇数2n+1有$\tan((2n+1)x)=(-1)^n\prod_{h=0}^{2n} \tan(x+\frac{h\pi}{2n+1})$
由此我们可以得到由此我们可以得到
$\tan(15\degree)=\tan(1\degree)\tan(11\degree)\tan(13\degree)\tan(23\degree)\tan(25\degree)...\tan(71\degree)\tan(73\degree)\tan(83\degree)\tan(85\degree)$
以及$\tan(30\degree)=\tan(2\degree)\tan(10\degree)\tan(14\degree)\tan(22\degree)\tan(26\degree)...\tan(70\degree)\tan(74\degree)\tan(82\degree)\tan(86\degree)$
两者相乘再添加$\tan(45\degree)$已经可以得到一个33项结果了。
如果我们再考虑用5倍角关系替换左边两项，
也就是$\tan(15\degree)=\tan(3\degree)\tan(39\degree)\tan(75\degree)\tan(111\degree)\tan(147\degree)=\tan(3\degree)\tan(39\degree)\tan(75\degree)\tan(69\degree)\tan(33\degree)$
$\tan(30\degree)=\tan(6\degree)\tan(42\degree)\tan(78\degree)\tan(114\degree)\tan(150\degree)=\tan(6\degree)\tan(42\degree)\tan(78\degree)\tan(66\degree)\tan(30\degree)$
应该可以达到41项:
$\tan(1\degree)\tan(11\degree)\tan(13\degree)\tan(23\degree)\tan(25\degree)...\tan(71\degree)\tan(73\degree)\tan(83\degree)\tan(85\degree)\times \tan(2\degree)\tan(10\degree)\tan(14\degree)\tan(22\degree)\tan(26\degree)...\tan(70\degree)\tan(74\degree)\tan(82\degree)\tan(86\degree)=\tan(3\degree)\tan(39\degree)\tan(75\degree)\tan(69\degree)\tan(33\degree)\times \tan(6\degree)\tan(42\degree)\tan(78\degree)\tan(66\degree)\tan(30\degree)\times \tan(45\degree)$

王守恩 · 发表于 2024-3-26 10:17:10

抛砖引玉, 列出一些不成熟恒等式。问：
尽量不用45。这些45能去掉吗？
尽量不用90-a。这些90-a能去掉吗？

1,  tan(45)=1

2,  tan(a)*tan(90-a)=1

3,  tan(a)*tan(90-a)*tan(45)=1

4,  tan(a)*tan(60-a)*tan(60+a)*tan(90-3a)=1

5,  tan(a)*tan(60-a)*tan(60+a)*tan(90-3a)*tan(45)=1

6,  tan(a)*tan(60-a)*tan(60+a)*tan(90-3a)*tan(b)*tan(90-b)=1

7,  tan(a)*tan(60-a)*tan(60+a)*tan(90-3a)*tan(b)*tan(90-b)*tan(45)=1

8,  tan(a)*tan(60-a)*tan(60+a)*tan(90-3a)*tan(b)*tan(60-b)*tan(60+b)*tan(90-3b)=1

9,  tan(a)*tan(60-a)*tan(60+a)*tan(90-3a)*tan(b)*tan(60-b)*tan(60+b)*tan(90-3b)*tan(45)=1
......

mathe · 发表于 2024-3-26 12:32:22

41项的方案很多，写了个程序去寻找，已经找到了8千多项候选解，比如

[41:{234236d9,c402220}]2,3,12,15,16,17,19,20,21,22,23,25,26,29,30,31,33,34,35,37,38,39,40,41,42,43,44,45,54,58,62,72,76,77,79,80,82,83,85,86,89,
[41:{23577698,c402225}]1,2,3,7,12,16,20,22,26,29,30,31,33,34,35,37,38,39,40,41,42,43,44,45,54,58,62,65,67,69,71,72,73,75,76,77,79,80,82,85,86,
[41:{234236dc,c402360}]1,2,12,15,16,17,19,20,21,22,23,25,26,30,33,34,35,37,38,39,40,41,42,43,44,45,54,58,59,61,62,72,76,77,79,80,82,83,85,86,87,
[41:{23423ed1,c402a28}]2,3,4,15,16,17,19,20,21,22,23,25,29,30,31,33,35,37,38,39,40,41,42,43,44,45,54,56,58,62,64,72,76,77,78,79,80,82,83,85,89,
[41:{23577e90,c402a2d}]1,2,3,4,7,16,20,22,29,30,31,33,35,37,38,39,40,41,42,43,44,45,54,56,58,62,64,65,67,69,71,72,73,75,76,77,78,79,80,82,85,
[41:{23423ed4,c402b68}]1,2,4,15,16,17,19,20,21,22,23,25,30,33,35,37,38,39,40,41,42,43,44,45,54,56,58,59,61,62,64,72,76,77,78,79,80,82,83,85,87,
[41:{234276c9,c403224}]2,3,5,12,16,17,19,20,21,22,23,26,29,30,31,33,34,37,38,39,40,41,42,43,44,45,54,55,58,62,65,72,75,76,77,79,80,82,83,86,89,
[41:{234276cc,c403364}]1,2,5,12,16,17,19,20,21,22,23,26,30,33,34,37,38,39,40,41,42,43,44,45,54,55,58,59,61,62,65,72,75,76,77,79,80,82,83,86,87,
[41:{23427ec1,c403a2c}]2,3,4,5,16,17,19,20,21,22,23,29,30,31,33,37,38,39,40,41,42,43,44,45,54,55,56,58,62,64,65,72,75,76,77,78,79,80,82,83,89,
[41:{23427ec4,c403b6c}]1,2,4,5,16,17,19,20,21,22,23,30,33,37,38,39,40,41,42,43,44,45,54,55,56,58,59,61,62,64,65,72,75,76,77,78,79,80,82,83,87,
[41:{23523699,c406221}]2,3,7,12,15,16,17,19,20,22,25,26,29,30,31,33,34,35,38,39,40,41,42,43,44,45,53,54,58,62,67,69,72,76,77,79,80,82,85,86,89,
[41:{2352369c,c406361}]1,2,7,12,15,16,17,19,20,22,25,26,30,33,34,35,38,39,40,41,42,43,44,45,53,54,58,59,61,62,67,69,72,76,77,79,80,82,85,86,87,
[41:{23523e91,c406a29}]2,3,4,7,15,16,17,19,20,22,25,29,30,31,33,35,38,39,40,41,42,43,44,45,53,54,56,58,62,64,67,69,72,76,77,78,79,80,82,85,89,
[41:{23523e94,c406b69}]1,2,4,7,15,16,17,19,20,22,25,30,33,35,38,39,40,41,42,43,44,45,53,54,56,58,59,61,62,64,67,69,72,76,77,78,79,80,82,85,87,

复制代码

tanl4.zip (164.55 KB, 下载次数: 2)

由于使用近似计算，程序开始也找到一些42项的解，比如

[42:{15a524d,3700a0e9}] 2, 6, 8, 9, 11, 12, 14, 16, 17, 19, 22, 24, 25, 27, 32, 33, 34, 35, 37, 39, 40, 41, 42, 43, 45, 52, 54, 60, 61, 62, 64, 67, 69, 70, 72, 75, 77, 80, 83, 86, 87, 89,
[42:{14a134d,3700a4e9}] 2, 6, 8, 11, 12, 14, 15, 16, 17, 19, 21, 22, 24, 25, 27, 32, 34, 35, 37, 39, 40, 41, 42, 43, 45, 52, 54, 57, 60, 61, 62, 64, 67, 70, 72, 77, 80, 81, 83, 86, 87, 89,
[42:{15a5349,3700a4f9}] 2, 3, 6, 8, 11, 12, 14, 16, 17, 19, 22, 24, 25, 32, 34, 35, 37, 39, 40, 41, 42, 43, 45, 52, 54, 57, 60, 61, 62, 63, 64, 67, 69, 70, 72, 75, 77, 80, 81, 83, 86, 89,
[42:{15a124d,3701a4f9}] 2, 6, 8, 9, 11, 12, 14, 15, 16, 17, 19, 22, 24, 25, 32, 34, 35, 37, 40, 41, 42, 43, 45, 51, 52, 54, 57, 60, 61, 62, 63, 64, 67, 69, 70, 72, 77, 80, 83, 86, 87, 89,
[42:{15b460d,3701b0ad}] 2, 6, 7, 8, 9, 12, 13, 14, 16, 19, 22, 24, 27, 29, 32, 33, 34, 37, 40, 41, 42, 43, 45, 51, 52, 54, 55, 60, 62, 64, 65, 67, 69, 70, 72, 73, 75, 79, 80, 86, 87, 89,
[42:{14b070d,3701b4ad}] 2, 6, 7, 8, 12, 13, 14, 15, 16, 19, 21, 22, 24, 27, 29, 32, 34, 37, 40, 41, 42, 43, 45, 51, 52, 54, 55, 57, 60, 62, 64, 65, 67, 70, 72, 73, 79, 80, 81, 86, 87, 89,
[42:{15b4709,3701b4bd}] 2, 3, 6, 7, 8, 12, 13, 14, 16, 19, 22, 24, 29, 32, 34, 37, 40, 41, 42, 43, 45, 51, 52, 54, 55, 57, 60, 62, 63, 64, 65, 67, 69, 70, 72, 73, 75, 79, 80, 81, 86, 89,
[42:{15f420d,3705b4ad}] 2, 6, 7, 8, 9, 11, 12, 13, 14, 16, 22, 24, 27, 29, 32, 34, 37, 40, 42, 43, 45, 49, 51, 52, 54, 55, 57, 60, 62, 64, 65, 67, 69, 70, 71, 72, 73, 75, 80, 86, 87, 89,
[42:{164bca6,370e5a06}] 1, 4, 7, 9, 10, 15, 17, 18, 20, 21, 23, 26, 27, 28, 29, 30, 33, 36, 38, 39, 43, 45, 48, 49, 50, 53, 55, 56, 58, 65, 66, 68, 71, 74, 76, 77, 78, 79, 82, 84, 87, 88,
[42:{174fca2,370e5a16}] 1, 3, 4, 7, 9, 10, 17, 18, 20, 23, 26, 28, 29, 30, 33, 36, 38, 39, 43, 45, 48, 49, 50, 53, 55, 56, 58, 63, 65, 66, 68, 69, 71, 74, 75, 76, 77, 78, 79, 82, 84, 88,
[42:{164bda2,370e5e16}] 1, 3, 4, 7, 10, 15, 17, 18, 20, 21, 23, 26, 28, 29, 30, 36, 38, 39, 43, 45, 48, 49, 50, 53, 55, 56, 57, 58, 63, 65, 66, 68, 71, 74, 76, 77, 78, 79, 81, 82, 84, 88,
[42:{15b424d,3711a0e9}] 2, 6, 8, 9, 11, 12, 13, 14, 16, 19, 22, 24, 25, 27, 32, 33, 34, 35, 37, 40, 41, 42, 45, 47, 51, 52, 54, 60, 61, 62, 64, 67, 69, 70, 72, 73, 75, 80, 83, 86, 87, 89,
[42:{14b034d,3711a4e9}] 2, 6, 8, 11, 12, 13, 14, 15, 16, 19, 21, 22, 24, 25, 27, 32, 34, 35, 37, 40, 41, 42, 45, 47, 51, 52, 54, 57, 60, 61, 62, 64, 67, 70, 72, 73, 80, 81, 83, 86, 87, 89,
[42:{15b4349,3711a4f9}] 2, 3, 6, 8, 11, 12, 13, 14, 16, 19, 22, 24, 25, 32, 34, 35, 37, 40, 41, 42, 45, 47, 51, 52, 54, 57, 60, 61, 62, 63, 64, 67, 69, 70, 72, 73, 75, 80, 81, 83, 86, 89,
[42:{160bde2,371a4a52}] 1, 3, 4, 10, 15, 17, 18, 19, 20, 21, 23, 25, 26, 28, 30, 33, 35, 36, 38, 39, 41, 45, 47, 48, 50, 53, 56, 58, 61, 63, 66, 68, 74, 76, 77, 78, 79, 81, 82, 83, 84, 88,
[42:{164b8e6,371e4a42}] 1, 4, 9, 10, 11, 15, 17, 18, 20, 21, 23, 25, 26, 27, 28, 30, 33, 35, 36, 38, 39, 45, 47, 48, 49, 50, 53, 56, 58, 61, 66, 68, 71, 74, 76, 77, 78, 82, 83, 84, 87, 88,
[42:{174f8e2,371e4a52}] 1, 3, 4, 9, 10, 11, 17, 18, 20, 23, 25, 26, 28, 30, 33, 35, 36, 38, 39, 45, 47, 48, 49, 50, 53, 56, 58, 61, 63, 66, 68, 69, 71, 74, 75, 76, 77, 78, 82, 83, 84, 88,
[42:{164b9e2,371e4e52}] 1, 3, 4, 10, 11, 15, 17, 18, 20, 21, 23, 25, 26, 28, 30, 35, 36, 38, 39, 45, 47, 48, 49, 50, 53, 56, 57, 58, 61, 63, 66, 68, 71, 74, 76, 77, 78, 81, 82, 83, 84, 88,
[42:{165eda2,371e5a06}] 1, 3, 4, 7, 10, 13, 18, 20, 21, 23, 26, 27, 28, 29, 30, 33, 36, 38, 39, 45, 47, 48, 49, 50, 53, 55, 56, 58, 65, 66, 68, 71, 73, 74, 75, 76, 78, 79, 81, 82, 84, 88,
[42:{165aca6,371f5a06}] 1, 4, 7, 9, 10, 13, 15, 18, 20, 21, 23, 26, 27, 28, 29, 30, 33, 36, 38, 45, 47, 48, 49, 50, 51, 53, 55, 56, 58, 65, 66, 68, 71, 73, 74, 76, 78, 79, 82, 84, 87, 88,
[42:{175eca2,371f5a16}] 1, 3, 4, 7, 9, 10, 13, 18, 20, 23, 26, 28, 29, 30, 33, 36, 38, 45, 47, 48, 49, 50, 51, 53, 55, 56, 58, 63, 65, 66, 68, 69, 71, 73, 74, 75, 76, 78, 79, 82, 84, 88,
[42:{165ada2,371f5e16}] 1, 3, 4, 7, 10, 13, 15, 18, 20, 21, 23, 26, 28, 29, 30, 36, 38, 45, 47, 48, 49, 50, 51, 53, 55, 56, 57, 58, 63, 65, 66, 68, 71, 73, 74, 76, 78, 79, 81, 82, 84, 88,

复制代码

但是提高计算精度验算以后，全部淘汰。
现在程序已经能够证明最多只能41项了，但是41项的穷举还没有完成。

王守恩 · 发表于 2024-3-26 14:53:35

mathe 发表于 2024-3-23 19:06
上面代码通过浮点计算找出了38组解，但是并没有证明它们的正切值乘积严格等于1。
如果我们记\(s_1(x)=1, c_ ...

谢谢mathe！4个数(主帖)有38组解。5个数也是38组解。6个数能来几个？我只找到1个:{6,30,42,60,66,78}。谢谢mathe！

王守恩 · 发表于 2024-3-26 17:20:09

northwolves 发表于 2024-3-23 23:50
{38,{{1,59,61,87},{2,58,62,84},{3,29,31,89},{3,39,75,81},{3,57,63,81},{3,63,69,75},{4,56,64,78},{5 ...

38=19*2, 每个解倒过来就是另外1个解。
{19, {{1, 59, 61, 87}, {2, 58, 62, 84}, {3, 39, 75, 81}, {3, 57, 63, 81}, {3, 63, 69, 75}, {4, 56, 64, 78}, {5, 55, 65, 75}, {6, 42, 66, 78}, {6, 54, 66, 72}, {7, 53, 67, 69},
{8, 52, 66, 68}, {9, 33, 69, 75}, {9, 51, 63, 69}, {10, 50, 60, 70}, {11, 49, 57, 71}, {12, 48, 54, 72}, {13, 47, 51, 73}, {14, 46, 48, 74}, {15, 51, 57, 63}}}

v=Select[Subsets[Range@87,{4}],N[Times@@Tan[#Degree]]==1&&#[[4]]<90-#[[1]]&];{Length@v,v}

复制代码

王守恩 · 发表于 2024-3-27 01:29:52

赌一把。
0, 1项有解, 2, 3项无解;
4, 5项有解, 6, 7项无解;
8, 9项有解, 10, 11项无解;
......
40, 41项有解, 42, 43项无解;
......
11个有解在慢慢长大: 1, 38, ?, ?, ?, ?, ?, ?, ?, ?, 8360。

王守恩 · 发表于 2024-3-27 07:32:45

mathe 发表于 2024-3-26 12:32
41项的方案很多，写了个程序去寻找，已经找到了8千多项候选解，比如

试试8项的方案不会超过140。

账号		自动登录	找回密码
密码			欢迎注册

[讨论] 这样的数组有多少？

点评

点评

评分

点评

点评