这样的数组有多少？

王守恩 · 发表于 2024-3-23 12:11:37

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a,b,c,d=正整数, a<b<c<d<90, 满足 $\tan(a^\circ)*\tan(b^\circ)*\tan(c^\circ)*\tan(d^\circ)$=1, 这样的数组有多少？

补充内容 (2024-3-24 06:32):
不好意思，还得限制：a+d≠90。

mathe · 发表于 2024-3-23 19:06:13

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$s_1(x)=1, c_1(x)=x$,于是$\sin(x)=\sin(x) s_1(\cos(x)),\cos(x)=c_1(\cos(x))$
假设已经存在整系数多项式$s_n(x),c_n(x)$使得$\sin(nx)=\sin(x) s_n(\cos(x)), \cos(nx)=c_n(\cos(nx))$
于是我们可以得到$sin((n+1)x) = \sin(x) s_n(\cos(x))\cos(x)+c_n(x)\sin(x), \cos((n+1)x)=c_n(\cos(x))cos(x)-s_n(\cos(x))\sin^2(x)=c_n(\cos(x))cos(x)-s_n(\cos(x))(1-\cos^2(x))$
于是我们可以得到$s_{n+1}(x)=x*s_n(x)+c_n(x), c_{n+1}(x)=x*c_n(x)-s_n(x)(1-x^2)$. 所以我们知道$s_n,c_n$都是整系数多项式。
如果$\tan(a\degree)\tan(b\degree)\tan(c\degree)\tan(d\degree)=1$，那么$\sin(a\degree)\sin(b\degree)\sin(c\degree)\sin(d\degree)=\cos(a\degree)\cos(b\degree)\cos(c\degree)\cos(d\degree)$
于是设$u=\cos(1\degree),v=\sin(1\degree)$,那么必然有$v^4 s_a(u)s_b(u)s_c(u)s_d(u)=c_a(u)c_b(u)c_c(u)c_d(u)$,故$(1-u^2)^2 s_a(u)s_b(u)s_c(u)s_d(u)-c_a(u)c_b(u)c_c(u)c_d(u)=0$, 我们可以得到一个以u为零点的整系数多项式，这个多项式必然是u的极小多项式的倍数。
另外$c_{90}(u)=0$, 当然，对于1到180之间的任意一个奇数h都有$c_{90}(\cos(h\degree))=0$,而如果h含因子3或5,显然$\cos(h\degree)$可以有次数更低的多项式作为极小多项式。
由此，我们可以知道u的极小多项式的次数很可能是1到180之间和90互素的整数数目。
计算$c_{90}(x)$并且因子分解发现它正好有一个48次不可约因式$m(x)=281474976710656*x^48 - 3377699720527872*x^46 + 18999560927969280*x^44 - 66568831992070144*x^42 + 162828875980603392*x^40 - 295364007592722432*x^38 + 411985976135516160*x^36 - 452180272956309504*x^34 + 396366279591591936*x^32 - 280058255978266624*x^30 + 160303703377575936*x^28 - 74448984852135936*x^26 + 28011510450094080*x^24 - 8500299631165440*x^22 + 2064791072931840*x^20 - 397107008634880*x^18 + 59570604933120*x^16 - 6832518856704*x^14 + 583456329728*x^12 - 35782471680*x^10 + 1497954816*x^8 - 39625728*x^6 + 579456*x^4 - 3456*x^2 + 1$
这个验证了这个多项式的48个根正好都是$\cos(h\degree)$其中h为和90互素的整数。
而如果一组a,b,c,d满足要求，那么必然$m(x)|(1-x^2)^2 s_a(x)s_b(x)s_c(x)s_d(x)-c_a(x)c_b(x)c_c(x)c_d(x)$。
于是，如果a,b,c,d满足要求，那么我们将它们都乘上一个和90互素的整数h以后，结果也会满足要求，只是乘上h以后这些角度可能会超过90度，而正切函数以180度为周期，大于180度的可以减去180的倍数，90度到180度之间的可以变换为用180度减这个角，是的正切值变号，由于最终乘积是正数1，所以变号的数目必然会正好是偶数个。
于是，这38组解中，如果任意两组解之间可以通过上面乘h的方法相互变换的，我们就认为它们等价，这样每个等价组中只需要验证一组即可。
我们可以分析出38组等价关系如下

{0}:1 59 61 87 =>{0}
{1}:2 58 62 84 =>{1}
{2}:3 29 31 89 =>{0}
{3}:3 39 75 81 =>{3}
{4}:3 57 63 81 =>{4}
{5}:3 63 69 75 =>{3}
{6}:4 56 64 78 =>{6}
{7}:5 55 65 75 =>{7}
{8}:6 28 32 88 =>{6}
{9}:6 42 66 78 =>{9}
{10}:6 54 66 72 =>{10}
{11}:7 53 67 69 =>{0}
{12}:8 52 66 68 =>{6}
{13}:9 15 51 87 =>{3}
{14}:9 27 33 87 =>{4}
{15}:9 33 69 75 =>{3}
{16}:9 51 63 69 =>{4}
{17}:10 50 60 70 =>{17}
{18}:11 49 57 71 =>{0}
{19}:12 24 48 84 =>{19}
{20}:12 26 34 86 =>{1}
{21}:12 48 54 72 =>{21}
{22}:13 47 51 73 =>{0}
{23}:14 46 48 74 =>{1}
{24}:15 21 27 87 =>{3}
{25}:15 21 57 81 =>{3}
{26}:15 25 35 85 =>{7}
{27}:15 51 57 63 =>{3}
{28}:16 42 44 76 =>{6}
{29}:17 39 43 77 =>{0}
{30}:18 24 36 84 =>{21}
{31}:18 36 42 78 =>{10}
{32}:19 33 41 79 =>{0}
{33}:20 30 40 80 =>{33}
{34}:21 23 37 83 =>{0}
{35}:21 27 39 81 =>{4}
{36}:22 24 38 82 =>{1}
{37}:27 33 39 75 =>{3}

复制代码

最后还余12组需要验证。
使用pari/gp计算上面过程验证全部通过，输出结果如下

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Verify passed for [1, 59, 61, 87]
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Verify passed for [2, 58, 62, 84]
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Verify passed for [3, 39, 75, 81]
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Verify passed for [3, 57, 63, 81]
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Verify passed for [4, 56, 64, 78]
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Verify passed for [5, 55, 65, 75]
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Verify passed for [6, 42, 66, 78]
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Verify passed for [6, 54, 66, 72]
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Verify passed for [10, 50, 60, 70]
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Verify passed for [12, 24, 48, 84]
Pol for [12, 48, 54, 72]is -730750818665451459101842416358141509827966271488*x^162 + 29595408155950784093624617862504731148032633995264*x^160 - 588208737099521833860789280017281531567148600655872*x^158 + 7647330153297032814858877819763462092271849155067904*x^156 - 73145744899811017983895437374161595486998936934154240*x^154 + 548872624627180638769600094079616405370913678994440192*x^152 - 3364776837725173616730732200671836467968272104775745536*x^150 + 17327825420082403302749277577192521903016332152474173440*x^148 - 76498411087522882762705617485673804083202784559502458880*x^146 + 294021670792095030516789746218539294197785575367586611200*x^144 - 995804974498490274408180008903315977980394830258115706880*x^142 + 3000904154035034959994066848022605736171502903586392637440*x^140 - 8110777060766913933539519564238876059152423125526555656192*x^138 + 19791217231654526952439101301174521266692786733506073985024*x^136 - 43842513075914227177459206067756299524189136634552548065280*x^134 + 88589712929585414598262808451196459197290112596484751884288*x^132 - 163944062069605200046357072317860570680601522583306328080384*x^130 + 278837923012908641458154979997010321532828350276069687820288*x^128 - 437190323736288857594884968637287788082335808457541362384896*x^126 + 633580014615987256819970395292352103634227915274777001984000*x^124 - 850648097088295566991037717341457243365250366529128418508800*x^122 + 1060221817731967169868431170499374520985969766185663096422400*x^120 - 1228893470552961946892954311260638649324646774442473134489600*x^118 + 1326720617332983471911489460530526837079768014384797004595200*x^116 - 1335933954953351412688652581784211051226155292262469206016000*x^114 + 1256167971365625758996731573616365457072652005468987468021760*x^112 - 1104120490668655220294638498529316696997116072680320432865280*x^110 + 907983860296102612574260329037483346844679129728192180387840*x^108 - 699137892429488392251342774677315935243677506872267379834880*x^106 + 504368987732707143230324415488626287222072248605673835724800*x^104 - 341085896249542861002729491080439696257754919637877407088640*x^102 + 216317834471976763977475942939589371512397881193635376005120*x^100 - 128698711614938098760637730474875888099323258642847970099200*x^98 + 71846955168639979258786837320540885146356994166015551078400*x^96 - 37640408498276459721974721758003956372631881134769176576000*x^94 + 18506887013044590038779697278316680911673110510694721126400*x^92 - 8539443235957303119745570194778993619959015676697513033728*x^90 + 3697348125406378269684433365418535728436564785177036849152*x^88 - 1501851509343762310733447678604165211647751001994694754304*x^86 + 572156271279075593676334299322844402112498505651983482880*x^84 - 204358274762178024544641534797742976300895103674417676288*x^82 + 68400910147671156975685807768274228565165328157040967680*x^80 - 21443142466928656353488819624239056658989683719805075456*x^78 + 6292139615808750735736497076571570353122799107610509312*x^76 - 1726945098714843951930841161033005374987813255957708800*x^74 + 442970437951452375514411615043072727610011614699323392*x^72 - 106093744321955797584081328437595033292085296822747136*x^70 + 23701793944266720737187995540144176421776300890390528*x^68 - 4933542270197330943302943834891494101273638973997056*x^66 + 955612079607483734682816217261380546415068425748480*x^64 - 172010174329347072622930466447515256559068912812032*x^62 + 28729132631760759398265007170031070288994514239488*x^60 - 4444979436907040804966167556944748602486482272256*x^58 + 635931366557671822006057162965534055944696102912*x^56 - 83962131730008201646671750637419512475020165120*x^54 + 10208140229876151046400163580171483179273682944*x^52 - 1140161215028305434121576559107689260859260928*x^50 + 116683165280683921609789855267171435932549120*x^48 - 10910030705956054670508807063696977417994240*x^46 + 929067120345462698669534888986238622105600*x^44 - 71805350968223952310854009835239734509568*x^42 + 5017400355848762966114180350422357639168*x^40 - 315610667566660081706539171554879602688*x^38 + 17786940785161481819731785098624761856*x^36 - 893317339333830094193397278567301120*x^34 + 39742349639045484349168296602370048*x^32 - 1555571783436726970410350214643712*x^30 + 53155832830432036472980531838976*x^28 - 1571723051353688448509707026432*x^26 + 39801326561000669963721113600*x^24 - 852886897567872424306802688*x^22 + 15246704496160437202059264*x^20 - 223528225321647891382272*x^18 + 2632130050303785222144*x^16 - 24256824985682565120*x^14 + 169266650837480448*x^12 - 856698182226432*x^10 + 2970010872576*x^8 - 6535323952*x^6 + 8274120*x^4 - 5274*x^2 + 1
Verify passed for [12, 48, 54, 72]
Pol for [20, 30, 40, 80]is -170141183460469231731687303715884105728*x^130 + 5529588462465250031279837370766233436160*x^128 - 87782216841635844246567418260913955799040*x^126 + 907198107123205083256848318641335173120000*x^124 - 6863385679485676552496751267816291696640000*x^122 + 40526919760227022907182816886201639210319872*x^120 - 194463848849476440562691742316854639759196160*x^118 + 779549331363493536123264284095962588999843840*x^116 - 2663726455017675146486768839610691838436966400*x^114 + 7877438979232711486924370108325282888600780800*x^112 - 20402566956212885010410947793925874259486310400*x^110 + 46720163828306705404768663142310718233116672000*x^108 - 95321012217549215467999408097741965197574144000*x^106 + 174379169293853705385701592067368825546342400000*x^104 - 287553827200037672671163052731458837765160960000*x^102 + 429330366361120071113740888816421713281053360128*x^100 - 582561763607491901232280720347080205861598003200*x^98 + 720696528612532686443472132286371341353667788800*x^96 - 815073455326990818366895263560361431273570304000*x^94 + 844638709486243040080387865087662320661823488000*x^92 - 803558555677505607135871397768118451052153405440*x^90 + 702982094471820168048639329599615900878005862400*x^88 - 566291136111557662916471793358040861427472793600*x^86 + 420518921812188303329423047590352291064971264000*x^84 - 288114976653860210000135363388770277453201408000*x^82 + 182253311203578287033286195909711810258008014848*x^80 - 106494177532700527478144440773415641489038376960*x^78 + 57497673746080613829832141261621047224162058240*x^76 - 28688447123684908502918643128551139948101632000*x^74 + 13227641987672668661121851213833847076552704000*x^72 - 5634979164869772241311461094039500602227032064*x^70 + 2217088455693549101452254097934867619769221120*x^68 - 805251258029759877121078990446862103614586880*x^66 + 269801381513421365883086559046821802200268800*x^64 - 83321351231743541720936886533627506694553600*x^62 + 23693481525072062920382416562715890634719232*x^60 - 6196492048537491186266719043361832414740480*x^58 + 1488367823662227817687803946076160833617920*x^56 - 327822570675186625307237226793182586470400*x^54 + 66093126739663162580411460630734490828800*x^52 - 12172723106768305860324652904396789645312*x^50 + 2043383985562338213862753120410822246400*x^48 - 311846591182153854365506482531309977600*x^46 + 43144700632732534020264920233279488000*x^44 - 5394234846293467808306244706369536000*x^42 + 607302005688321703458356901760204800*x^40 - 61322994504881012598141750345728000*x^38 + 5528994272077739970147513270272000*x^36 - 442885385699804057343969198080000*x^34 + 31339856182460490079126159360000*x^32 - 1946586255582214147116476923904*x^30 + 105352626070891779801250529280*x^28 - 4926911265917821326270136320*x^26 + 197181542837300315553792000*x^24 - 6677889274843392180224000*x^22 + 188867361951399328612352*x^20 - 4391279333204453621760*x^18 + 82354253920253706240*x^16 - 1216916135761254400*x^14 + 13753959338316800*x^12 - 114361530639616*x^10 + 662913895040*x^8 - 2472174160*x^6 + 5182200*x^4 - 4650*x^2 + 1
Verify passed for [20, 30, 40, 80]

复制代码

mathe · 发表于 2024-4-2 12:12:52

说实在，看不大明白需求。
不过由于解太多了，我们不妨多加一些约束。
要求$/_PAB, /_PAD, /_PDA,/_PDC,/_PCD,/_PCB,/_PBC,/_PBA$全部两两不等，而且要求四边形为凸四边形，这样的约束下有1092组不同的解

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3 6 93 48 30 12 18 150
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3 30 87 81 12 9 132 6
3 30 93 48 18 12 150 6
6 3 30 126 15 18 75 87
6 3 30 132 9 12 81 87
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6 3 150 18 12 30 48 93
6 3 150 18 48 30 12 93
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12 9 48 15 18 105 54 99
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27 24 63 9 48 81 12 96
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27 24 63 15 54 75 18 84
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27 24 63 21 30 69 42 84
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27 24 63 48 81 12 9 96
27 24 63 54 75 18 15 84
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27 24 117 12 30 48 18 84
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30 12 24 15 54 105 18 102
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30 18 36 9 48 99 12 108
30 18 36 12 99 48 9 108
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30 18 36 48 99 12 9 108
30 18 36 99 48 9 12 108
30 18 144 12 54 24 6 72
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30 21 42 24 63 84 27 69
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30 24 54 9 48 99 12 84
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30 36 72 18 96 42 12 54
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30 36 72 24 78 48 18 54
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30 36 72 42 96 18 12 54
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30 51 78 54 75 18 15 39
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30 54 72 9 48 99 12 36
30 54 72 12 42 96 18 36
30 54 72 12 99 48 9 36
30 54 72 12 126 24 6 36
30 54 72 18 48 78 24 36
30 54 72 18 96 42 12 36
30 54 72 24 78 48 18 36
30 54 72 24 126 12 6 36
30 54 72 42 96 18 12 36
30 54 72 48 78 24 18 36
30 54 72 48 99 12 9 36
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30 54 72 99 48 9 12 36
30 66 48 15 54 105 18 24
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30 69 42 24 63 84 27 21
30 69 42 27 84 63 24 21
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30 84 54 9 48 99 12 24
30 84 54 12 42 96 18 24
30 84 54 12 99 48 9 24
30 84 54 18 96 42 12 24
30 84 54 42 96 18 12 24
30 84 54 48 99 12 9 24
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30 84 54 99 48 9 12 24
30 96 54 12 42 84 18 24
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30 96 54 42 84 18 12 24
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30 102 24 15 54 105 18 12
30 102 24 18 105 54 15 12
30 102 24 54 105 18 15 12
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30 108 36 9 48 99 12 18
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30 108 36 12 126 24 6 18
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36 18 30 9 48 99 12 108
36 18 30 12 99 48 9 108
36 18 30 12 126 24 6 108
36 18 30 24 126 12 6 108
36 18 30 48 99 12 9 108
36 18 30 99 48 9 12 108
36 30 54 9 48 99 12 72
36 30 54 12 42 96 18 72
36 30 54 12 99 48 9 72
36 30 54 12 126 24 6 72
36 30 54 18 48 78 24 72
36 30 54 18 96 42 12 72
36 30 54 24 78 48 18 72
36 30 54 24 126 12 6 72
36 30 54 42 96 18 12 72
36 30 54 48 78 24 18 72
36 30 54 48 99 12 9 72
36 30 54 78 48 18 24 72
36 30 54 96 42 12 18 72
36 30 54 99 48 9 12 72
36 30 126 12 54 24 6 72
36 30 126 24 54 12 6 72
36 72 54 9 48 99 12 30
36 72 54 12 42 96 18 30
36 72 54 12 99 48 9 30
36 72 54 12 126 24 6 30
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36 72 54 18 96 42 12 30
36 72 54 24 78 48 18 30
36 72 54 24 126 12 6 30
36 72 54 42 96 18 12 30
36 72 54 48 78 24 18 30
36 72 54 48 99 12 9 30
36 72 54 78 48 18 24 30
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36 72 54 99 48 9 12 30
36 108 30 9 48 99 12 18
36 108 30 12 99 48 9 18
36 108 30 12 126 24 6 18
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36 108 30 48 99 12 9 18
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39 30 51 15 54 75 18 78
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39 30 51 54 75 18 15 78
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42 21 30 24 63 84 27 69
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42 21 30 63 84 27 24 69
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42 84 18 24 54 96 30 12
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42 96 18 84 54 24 30 12
48 9 12 15 54 105 18 99
48 9 12 18 105 54 15 99
48 9 12 24 54 84 30 99
48 9 12 24 63 96 27 81
48 9 12 27 96 63 24 81
48 9 12 30 54 72 36 99
48 9 12 30 84 54 24 99
48 9 12 30 108 36 18 99
48 9 12 36 72 54 30 99
48 9 12 36 108 30 18 99
48 9 12 54 72 36 30 99
48 9 12 54 84 30 24 99
48 9 12 54 105 18 15 99
48 9 12 63 96 27 24 81
48 9 12 72 54 30 36 99
48 9 12 84 54 24 30 99
48 9 12 96 63 24 27 81
48 9 12 105 54 15 18 99
48 18 24 30 54 72 36 78
48 18 24 36 72 54 30 78
48 18 24 54 72 36 30 78
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48 24 30 15 54 105 18 66
48 24 30 18 105 54 15 66
48 24 30 54 105 18 15 66
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48 66 30 54 105 18 15 24
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48 81 12 24 63 96 27 9
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48 99 12 15 54 105 18 9
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48 99 12 24 54 84 30 9
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48 99 12 54 72 36 30 9
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48 99 12 54 105 18 15 9
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48 99 12 84 54 24 30 9
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51 30 39 15 54 75 18 78
51 30 39 18 75 54 15 78
51 30 39 54 75 18 15 78
51 30 39 75 54 15 18 78
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51 78 39 54 75 18 15 30
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54 15 18 12 99 48 9 105
54 15 18 12 126 24 6 105
54 15 18 24 63 84 27 75
54 15 18 24 102 30 12 105
54 15 18 24 126 12 6 105
54 15 18 27 84 63 24 75
54 15 18 30 66 48 24 105
54 15 18 30 102 24 12 105
54 15 18 39 78 51 30 75
54 15 18 48 66 30 24 105
54 15 18 48 99 12 9 105
54 15 18 51 78 39 30 75
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54 24 30 12 99 48 9 84
54 24 30 18 84 42 12 96
54 24 30 18 96 42 12 84
54 24 30 42 84 18 12 96
54 24 30 42 96 18 12 84
54 24 30 48 99 12 9 84
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54 30 36 12 126 24 6 72
54 30 36 18 96 42 12 72
54 30 36 24 78 48 18 72
54 30 36 24 126 12 6 72
54 30 36 42 96 18 12 72
54 30 36 48 78 24 18 72
54 30 36 48 99 12 9 72
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54 72 36 12 126 24 6 30
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54 72 36 24 126 12 6 30
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54 72 36 48 99 12 9 30
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54 75 18 63 84 27 24 15
54 75 18 84 63 24 27 15
54 84 30 12 99 48 9 24
54 84 30 18 96 42 12 24
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54 105 18 12 99 48 9 15
54 105 18 12 126 24 6 15
54 105 18 24 102 30 12 15
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54 105 18 30 66 48 24 15
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54 105 18 48 66 30 24 15
54 105 18 48 99 12 9 15
63 24 27 12 81 48 9 96
63 24 27 18 75 54 15 84
63 24 27 30 69 42 21 84
63 24 27 42 69 30 21 84
63 24 27 48 81 12 9 96
63 24 27 54 75 18 15 84
63 84 27 18 75 54 15 24
63 84 27 30 69 42 21 24
63 84 27 42 69 30 21 24
63 84 27 54 75 18 15 24
63 96 27 12 81 48 9 24
63 96 27 48 81 12 9 24
66 30 24 18 105 54 15 48
66 30 24 54 105 18 15 48
66 48 24 18 105 54 15 30
66 48 24 54 105 18 15 30
69 30 21 27 84 63 24 42
69 30 21 63 84 27 24 42
69 42 21 27 84 63 24 30
69 42 21 63 84 27 24 30
72 36 30 12 99 48 9 54
72 36 30 12 126 24 6 54
72 36 30 18 96 42 12 54
72 36 30 24 78 48 18 54
72 36 30 24 126 12 6 54
72 36 30 42 96 18 12 54
72 36 30 48 78 24 18 54
72 36 30 48 99 12 9 54
72 54 30 12 99 48 9 36
72 54 30 12 126 24 6 36
72 54 30 18 96 42 12 36
72 54 30 24 78 48 18 36
72 54 30 24 126 12 6 36
72 54 30 42 96 18 12 36
72 54 30 48 78 24 18 36
72 54 30 48 99 12 9 36
75 18 15 27 84 63 24 54
75 18 15 39 78 51 30 54
75 18 15 51 78 39 30 54
75 18 15 63 84 27 24 54
75 54 15 27 84 63 24 18
75 54 15 39 78 51 30 18
75 54 15 51 78 39 30 18
75 54 15 63 84 27 24 18
81 12 9 27 96 63 24 48
81 12 9 63 96 27 24 48
81 48 9 27 96 63 24 12
81 48 9 63 96 27 24 12
84 18 12 30 96 54 24 42
84 18 12 54 96 30 24 42
84 30 24 12 99 48 9 54
84 30 24 18 96 42 12 54
84 30 24 42 96 18 12 54
84 30 24 48 99 12 9 54
84 42 12 30 96 54 24 18
84 42 12 54 96 30 24 18
84 54 24 12 99 48 9 30
84 54 24 18 96 42 12 30
84 54 24 42 96 18 12 30
84 54 24 48 99 12 9 30
99 12 9 18 105 54 15 48
99 12 9 30 108 36 18 48
99 12 9 36 108 30 18 48
99 12 9 54 105 18 15 48
99 48 9 18 105 54 15 12
99 48 9 30 108 36 18 12
99 48 9 36 108 30 18 12
99 48 9 54 105 18 15 12
102 24 12 18 105 54 15 30
102 24 12 54 105 18 15 30
102 30 12 18 105 54 15 24
102 30 12 54 105 18 15 24
105 18 15 12 126 24 6 54
105 18 15 24 126 12 6 54
105 54 15 12 126 24 6 18
105 54 15 24 126 12 6 18
108 30 18 12 126 24 6 36
108 30 18 24 126 12 6 36
108 36 18 12 126 24 6 30
108 36 18 24 126 12 6 30

复制代码

比如随意选择了其中第790个作图，结果如下:

mathe · 发表于 2024-3-28 12:44:12

[6]10,16,20,52,56,88
[6]10,26,40,46,62,82
[6]11,25,35,47,61,83
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[6]12,16,24,44,76,84
[6]12,16,44,54,72,76
[6]12,24,28,32,48,88
[6]13,21,27,47,73,81
[6]13,23,25,49,59,85
[6]13,23,49,55,59,75
[6]13,27,33,47,73,75
[6]1,35,37,71,73,85
[6]1,37,65,71,73,75
[6]1,39,59,61,75,81
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[6]14,20,22,50,58,86
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[6]15,19,21,41,79,81
[6]15,19,41,51,63,79
[6]15,21,27,29,31,89
[6]15,31,35,41,67,77
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[6]18,26,34,36,42,86
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[6]25,27,33,35,39,85
[6]2,54,58,62,66,72
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[6]3,25,35,39,81,85
[6]3,25,35,63,69,85
[6]4,24,48,56,64,84
[6]4,32,40,68,70,76
[6]4,48,54,56,64,72
[6]5,11,47,61,75,83
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[6]5,21,27,55,65,87
[6]5,21,55,57,65,81
[6]5,31,41,65,67,77
[6]5,51,55,57,63,65
[6]5,9,51,55,65,87
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[6]9,23,37,51,63,83
[6]9,25,33,35,69,85
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[8]10,12,16,20,26,52,86,88
[8]10,12,24,48,50,60,70,84
[8]10,12,26,34,50,60,70,86
[8]10,12,48,50,54,60,70,72
[8]10,13,47,50,51,60,70,73
[8]10,14,20,46,48,52,56,88
[8]10,14,46,48,50,60,70,74
[8]10,15,21,27,50,60,70,87
[8]10,15,21,50,57,60,70,81
[8]10,15,25,35,50,60,70,85
[8]10,15,50,51,57,60,63,70
[8]10,16,20,22,24,56,82,88
[8]10,16,20,52,56,58,62,84
[8]10,16,26,40,42,62,76,82
[8]10,16,42,44,50,60,70,76
[8]10,17,39,43,50,60,70,77
[8]10,18,24,36,50,60,70,84
[8]10,18,36,42,50,60,70,78
[8]10,19,33,41,50,60,70,79
[8]10,21,23,37,50,60,70,83
[8]10,21,27,39,50,60,70,81
[8]10,22,24,38,50,60,70,82
[8]10,26,40,46,52,62,66,68
[8]10,27,33,39,50,60,70,75
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[8]11,12,26,34,49,57,71,86
[8]11,12,48,49,54,57,71,72
[8]11,13,25,27,47,49,53,89
[8]11,13,27,47,49,71,73,75
[8]11,13,47,49,51,57,71,73
[8]11,14,46,48,49,57,71,74
[8]1,11,49,57,59,61,71,87
[8]11,15,17,25,49,53,57,89
[8]11,15,21,27,49,57,71,87
[8]11,15,25,35,49,57,71,85
[8]11,15,31,35,57,67,71,77
[8]11,16,42,44,49,57,71,76
[8]11,17,25,27,39,49,53,89
[8]11,17,25,35,39,61,77,83
[8]11,17,39,43,49,57,71,77
[8]11,18,24,36,49,57,71,84
[8]11,18,36,42,49,57,71,78
[8]11,20,30,40,49,57,71,80
[8]11,21,23,37,49,57,71,83
[8]11,21,27,39,49,57,71,81
[8]11,22,24,38,49,57,71,82
[8]1,12,24,48,59,61,84,87
[8]1,12,26,34,59,61,86,87
[8]1,12,48,54,59,61,72,87
[8]11,25,27,35,39,49,71,85
[8]11,25,35,47,53,61,67,69
[8]11,27,31,35,39,67,71,77
[8]11,27,31,35,47,67,71,73
[8]1,13,47,51,59,61,73,87
[8]1,13,47,59,61,73,75,81
[8]1,14,46,48,59,61,74,87
[8]1,15,21,57,59,61,81,87
[8]1,15,25,35,59,61,85,87
[8]1,15,35,41,61,67,77,87
[8]1,15,51,57,59,61,63,87
[8]1,16,42,44,59,61,76,87
[8]1,17,39,43,59,61,77,87
[8]1,18,24,36,59,61,84,87
[8]1,18,36,42,59,61,78,87
[8]1,19,33,41,59,61,79,87
[8]1,19,41,59,61,63,79,81
[8]1,20,30,40,59,61,80,87
[8]1,21,23,37,59,61,83,87
[8]1,21,27,37,65,71,73,87
[8]1,21,27,39,59,61,81,87
[8]12,13,24,47,48,51,73,84
[8]12,13,26,34,47,51,73,86
[8]12,13,47,48,51,54,72,73
[8]1,21,37,41,65,73,79,81
[8]1,21,37,57,65,71,73,81
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[8]12,15,21,24,27,48,84,87
[8]12,15,21,24,48,57,81,84
[8]12,15,21,26,27,34,86,87
[8]12,15,21,26,34,57,81,86
[8]12,15,21,27,48,54,72,87
[8]12,15,21,48,54,57,72,81
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[8]12,15,24,48,51,57,63,84
[8]12,15,25,26,34,35,85,86
[8]12,15,25,35,48,54,72,85
[8]12,15,26,34,51,57,63,86
[8]12,15,48,51,54,57,63,72
[8]12,16,24,28,32,44,76,88
[8]12,16,26,34,42,44,76,86
[8]12,17,24,39,43,48,77,84
[8]12,17,26,34,39,43,77,86
[8]12,17,39,43,48,54,72,77
[8]12,18,24,26,34,36,84,86
[8]12,19,24,33,41,48,79,84
[8]12,19,26,33,34,41,79,86
[8]12,19,33,41,48,54,72,79
[8]12,20,24,30,40,48,80,84
[8]12,20,26,30,34,40,80,86
[8]12,20,30,40,48,54,72,80
[8]12,21,23,24,37,48,83,84
[8]12,21,23,26,34,37,83,86
[8]12,21,23,37,48,54,72,83
[8]12,21,24,27,39,48,81,84
[8]12,21,26,27,34,39,81,86
[8]12,21,27,39,48,54,72,81
[8]12,22,24,26,34,38,82,86
[8]12,22,24,38,48,54,72,82
[8]1,22,24,38,59,61,82,87
[8]12,24,27,33,39,48,75,84
[8]12,26,27,33,34,39,75,86
[8]12,26,32,34,40,68,70,76
[8]12,27,33,39,48,54,72,75
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[8]1,23,37,59,61,63,75,83
[8]1,25,35,39,59,61,81,85
[8]1,25,35,59,61,63,69,85
[8]1,2,58,59,61,62,84,87
[8]1,27,33,39,59,61,75,87
[8]13,14,46,47,48,51,73,74
[8]13,15,19,25,47,51,53,89
[8]13,15,21,27,47,51,73,87
[8]13,15,21,47,51,57,73,81
[8]13,15,25,35,47,51,73,85
[8]13,16,42,44,47,51,73,76
[8]13,18,24,36,47,51,73,84
[8]13,18,36,42,47,51,73,78
[8]13,19,21,23,55,59,79,81
[8]13,19,23,25,33,59,79,85
[8]13,19,23,33,55,59,75,79
[8]13,19,23,51,55,59,63,79
[8]13,19,25,27,33,47,53,89
[8]13,19,25,47,53,59,61,81
[8]13,19,33,41,47,51,73,79
[8]13,20,30,40,47,51,73,80
[8]13,21,23,27,29,49,55,89
[8]13,21,23,27,49,55,59,87
[8]13,21,23,37,47,51,73,83
[8]13,21,23,49,55,57,59,81
[8]13,22,24,38,47,51,73,82
[8]13,23,49,51,55,57,59,63
[8]13,25,27,33,35,47,73,85
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[8]14,21,27,39,46,48,74,81
[8]14,22,24,38,46,48,74,82
[8]14,27,33,39,46,48,74,75
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[8]15,17,21,27,39,43,77,87
[8]15,17,21,39,43,57,77,81
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[8]16,21,27,39,42,44,76,81
[8]16,22,24,38,42,44,76,82
[8]16,27,33,39,42,44,75,76
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[8]17,19,25,53,59,61,63,69
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[8]19,21,27,33,39,41,79,81
[8]19,22,24,33,38,41,79,82
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[8]20,27,30,33,39,40,75,80
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[8]21,22,23,24,37,38,82,83
[8]21,22,24,27,38,39,81,82
[8]2,12,26,34,58,62,84,86
[8]21,23,27,33,37,39,75,83
[8]21,23,29,37,43,55,65,79
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[8]21,27,29,31,37,65,71,73
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[8]27,31,33,35,39,41,67,77
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[8]3,13,47,51,63,69,73,75
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[8]3,14,46,48,63,69,74,75
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[8]3,15,29,31,51,57,63,89
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[8]3,16,42,44,57,63,76,81
[8]3,16,42,44,63,69,75,76
[8]3,17,19,23,25,63,83,89
[8]3,17,19,25,39,53,81,89
[8]3,17,19,25,53,63,69,89
[8]3,17,29,31,39,43,77,89
[8]3,17,39,43,57,63,77,81
[8]3,17,39,43,63,69,75,77
[8]3,18,24,29,31,36,84,89
[8]3,18,24,36,39,75,81,84
[8]3,18,24,36,57,63,81,84
[8]3,18,24,36,63,69,75,84
[8]3,18,29,31,36,42,78,89
[8]3,18,36,39,42,75,78,81
[8]3,18,36,42,57,63,78,81
[8]3,18,36,42,63,69,75,78
[8]3,19,29,31,33,41,79,89
[8]3,19,33,39,41,75,79,81
[8]3,19,33,41,63,69,75,79
[8]3,20,29,30,31,40,80,89
[8]3,20,30,39,40,75,80,81
[8]3,20,30,40,57,63,80,81
[8]3,20,30,40,63,69,75,80
[8]3,21,23,29,31,37,83,89
[8]3,21,23,37,39,75,81,83
[8]3,21,23,37,57,63,81,83
[8]3,21,27,29,31,39,81,89
[8]3,22,24,29,31,38,82,89
[8]3,22,24,38,39,75,81,82
[8]3,22,24,38,57,63,81,82
[8]3,22,24,38,63,69,75,82
[8]3,23,25,35,37,63,83,85
[8]3,23,29,37,43,63,79,85
[8]3,25,33,35,39,69,75,85
[8]3,27,29,31,33,39,75,89
[8]3,29,31,35,37,71,73,85
[8]3,29,31,37,65,71,73,75
[8]3,31,35,37,41,63,77,83
[8]3,31,35,39,41,67,77,81
[8]3,31,35,41,47,67,73,81
[8]3,31,35,41,63,67,69,77
[8]3,4,29,31,56,64,78,89
[8]3,4,39,56,64,75,78,81
[8]3,4,56,57,63,64,78,81
[8]3,4,56,63,64,69,75,78
[8]3,5,11,31,47,75,83,89
[8]3,5,29,31,55,65,75,89
[8]3,5,55,57,63,65,75,81
[8]3,6,28,29,31,32,88,89
[8]3,6,28,32,39,75,81,88
[8]3,6,28,32,57,63,81,88
[8]3,6,28,32,63,69,75,88
[8]3,6,29,31,42,66,78,89
[8]3,6,29,31,54,66,72,89
[8]3,6,39,42,66,75,78,81
[8]3,6,39,54,66,72,75,81
[8]3,6,42,57,63,66,78,81
[8]3,6,42,63,66,69,75,78
[8]3,6,54,57,63,66,72,81
[8]3,6,54,63,66,69,72,75
[8]3,7,13,29,73,79,81,85
[8]3,7,29,31,53,67,69,89
[8]3,7,29,39,43,79,81,85
[8]3,7,29,43,63,69,79,85
[8]3,7,39,53,67,69,75,81
[8]3,7,53,57,63,67,69,81
[8]3,8,29,31,52,66,68,89
[8]3,8,39,52,66,68,75,81
[8]3,8,52,57,63,66,68,81
[8]3,8,52,63,66,68,69,75
[8]3,9,29,31,33,69,75,89
[8]3,9,29,31,51,63,69,89
[8]4,10,40,46,56,62,78,82
[8]4,10,50,56,60,64,70,78
[8]4,11,49,56,57,64,71,78
[8]4,13,47,51,56,64,73,78
[8]4,14,46,48,56,64,74,78
[8]4,15,21,27,56,64,78,87
[8]4,15,21,56,57,64,78,81
[8]4,15,25,35,56,64,78,85
[8]4,15,51,56,57,63,64,78
[8]4,16,24,44,56,64,76,84
[8]4,16,42,44,56,64,76,78
[8]4,16,44,54,56,64,72,76
[8]4,17,39,43,56,64,77,78
[8]4,18,24,36,56,64,78,84
[8]4,19,33,41,56,64,78,79
[8]4,20,30,40,56,64,78,80
[8]4,21,23,37,56,64,78,83
[8]4,21,27,39,56,64,78,81
[8]4,22,24,38,56,64,78,82
[8]4,24,28,32,48,56,64,88
[8]4,24,32,38,40,70,76,82
[8]4,27,33,39,56,64,75,78
[8]4,32,40,46,48,68,70,74
[8]4,5,55,56,64,65,75,78
[8]4,6,28,32,56,64,78,88
[8]4,6,54,56,64,66,72,78
[8]4,7,53,56,64,67,69,78
[8]4,8,48,52,56,64,68,84
[8]4,8,52,56,64,66,68,78
[8]4,9,15,51,56,64,78,87
[8]4,9,27,33,56,64,78,87
[8]4,9,33,56,64,69,75,78
[8]4,9,51,56,63,64,69,78
[8]5,10,50,55,60,65,70,75
[8]5,11,17,39,61,75,77,83
[8]5,11,17,49,53,55,57,89
[8]5,11,17,57,61,63,77,83
[8]5,11,21,27,31,47,83,89
[8]5,11,21,27,47,61,83,87
[8]5,11,21,47,57,61,81,83
[8]5,11,27,31,47,53,67,89
[8]5,11,27,47,53,61,67,87
[8]5,11,31,57,65,67,71,77
[8]5,11,47,51,57,61,63,83
[8]5,11,47,53,57,61,67,81
[8]5,11,47,53,61,67,69,75
[8]5,11,49,55,57,65,71,75
[8]5,12,24,48,55,65,75,84
[8]5,12,26,34,55,65,75,86
[8]5,12,48,54,55,65,72,75
[8]5,13,19,47,51,53,55,89
[8]5,13,47,51,55,65,73,75
[8]5,14,46,48,55,65,74,75
[8]5,15,21,51,55,57,65,87
[8]5,16,42,44,55,65,75,76
[8]5,17,19,21,23,55,83,89
[8]5,17,19,41,61,63,77,83
[8]5,17,19,53,55,59,61,87
[8]5,17,39,43,55,65,75,77
[8]5,17,43,55,57,63,65,77
[8]5,18,24,36,55,65,75,84
[8]5,18,36,42,55,65,75,78
[8]5,19,21,41,47,61,81,83
[8]5,19,21,41,55,65,79,81
[8]5,19,33,41,47,61,75,83
[8]5,19,33,41,55,65,75,79
[8]5,19,41,47,51,61,63,83
[8]5,19,41,47,53,61,67,81
[8]5,19,41,51,55,63,65,79
[8]5,20,30,40,55,65,75,80
[8]5,21,23,37,55,65,75,83
[8]5,21,27,29,31,55,65,89
[8]5,21,27,39,55,65,75,81
[8]5,21,31,37,41,65,77,83
[8]5,22,24,38,55,65,75,82
[8]5,31,41,47,51,65,67,73
[8]5,6,28,32,55,65,75,88
[8]5,6,42,55,65,66,75,78
[8]5,6,54,55,65,66,72,75
[8]5,7,27,53,55,65,67,87
[8]5,7,53,55,57,65,67,81
[8]5,7,53,55,65,67,69,75
[8]5,8,52,55,65,66,68,75
[8]5,9,11,17,31,77,83,89
[8]5,9,11,17,61,77,83,87
[8]5,9,11,31,47,51,83,89
[8]5,9,11,47,51,61,83,87
[8]5,9,17,43,55,65,77,87
[8]5,9,27,33,55,65,75,87
[8]5,9,29,31,51,55,65,89
[8]5,9,51,55,63,65,69,75
[8]6,10,26,32,40,46,82,88
[8]6,10,28,32,50,60,70,88
[8]6,10,42,50,60,66,70,78
[8]6,10,50,54,60,66,70,72
[8]6,11,28,32,49,57,71,88
[8]6,11,42,49,57,66,71,78
[8]6,11,49,54,57,66,71,72
[8]6,12,26,28,32,34,86,88
[8]6,12,26,34,54,66,72,86
[8]6,12,28,32,48,54,72,88
[8]6,13,28,32,47,51,73,88
[8]6,13,42,47,51,66,73,78
[8]6,13,47,51,54,66,72,73
[8]6,14,20,22,28,50,86,88
[8]6,14,22,38,46,74,78,82
[8]6,14,26,34,46,66,74,86
[8]6,14,28,32,46,48,74,88
[8]6,14,46,48,54,66,72,74
[8]6,15,21,27,28,32,87,88
[8]6,15,21,27,42,66,78,87
[8]6,15,21,27,54,66,72,87
[8]6,15,21,28,32,57,81,88
[8]6,15,21,42,57,66,78,81
[8]6,15,21,54,57,66,72,81
[8]6,15,25,28,32,35,85,88
[8]6,15,25,35,42,66,78,85
[8]6,15,25,35,54,66,72,85
[8]6,15,28,32,51,57,63,88
[8]6,15,42,51,57,63,66,78
[8]6,15,51,54,57,63,66,72
[8]6,16,28,32,42,44,76,88
[8]6,16,42,44,54,66,72,76
[8]6,17,28,32,39,43,77,88
[8]6,17,39,42,43,66,77,78
[8]6,17,39,43,54,66,72,77
[8]6,18,28,32,36,42,78,88
[8]6,19,28,32,33,41,79,88
[8]6,19,33,41,42,66,78,79
[8]6,19,33,41,54,66,72,79
[8]6,20,28,30,32,40,80,88
[8]6,20,30,40,42,66,78,80
[8]6,20,30,40,54,66,72,80
[8]6,21,23,28,32,37,83,88
[8]6,21,23,37,42,66,78,83
[8]6,21,23,37,54,66,72,83
[8]6,21,27,28,32,39,81,88
[8]6,21,27,39,42,66,78,81
[8]6,21,27,39,54,66,72,81
[8]6,22,24,28,32,38,82,88
[8]6,22,26,34,38,42,82,86
[8]6,27,28,32,33,39,75,88
[8]6,27,33,39,42,66,75,78
[8]6,27,33,39,54,66,72,75
[8]6,28,32,34,38,70,74,80
[8]6,7,28,32,53,67,69,88
[8]6,7,42,53,66,67,69,78
[8]6,7,53,54,66,67,69,72
[8]6,8,28,32,52,66,68,88
[8]6,9,15,28,32,51,87,88
[8]6,9,15,42,51,66,78,87
[8]6,9,15,51,54,66,72,87
[8]6,9,27,28,32,33,87,88
[8]6,9,27,33,42,66,78,87
[8]6,9,27,33,54,66,72,87
[8]6,9,28,32,33,69,75,88
[8]6,9,28,32,51,63,69,88
[8]6,9,33,42,66,69,75,78
[8]6,9,33,54,66,69,72,75
[8]6,9,42,51,63,66,69,78
[8]6,9,51,54,63,66,69,72
[8]7,10,50,53,60,67,69,70
[8]7,11,49,53,57,67,69,71
[8]7,12,24,48,53,67,69,84
[8]7,12,26,34,53,67,69,86
[8]7,12,48,53,54,67,69,72
[8]7,13,15,29,51,73,79,85
[8]7,13,19,53,55,59,79,81
[8]7,13,25,49,53,59,69,85
[8]7,13,27,29,33,73,79,85
[8]7,13,27,29,49,53,55,89
[8]7,13,27,29,49,71,73,85
[8]7,13,27,47,53,67,73,81
[8]7,13,27,49,53,55,59,87
[8]7,13,29,51,55,65,73,79
[8]7,13,47,51,53,67,69,73
[8]7,13,49,53,55,57,59,81
[8]7,13,49,53,55,59,69,75
[8]7,14,46,48,53,67,69,74
[8]7,15,19,41,53,67,79,81
[8]7,15,25,35,53,67,69,85
[8]7,15,27,29,31,53,67,89
[8]7,15,29,43,49,57,71,85
[8]7,15,51,53,57,63,67,69
[8]7,16,42,44,53,67,69,76
[8]7,17,39,43,53,67,69,77
[8]7,18,24,36,53,67,69,84
[8]7,18,36,42,53,67,69,78
[8]7,19,33,41,53,67,69,79
[8]7,20,30,40,53,67,69,80
[8]7,22,24,38,53,67,69,82
[8]7,27,29,31,65,67,71,73
[8]7,27,29,33,39,43,79,85
[8]7,27,29,39,43,49,71,85
[8]7,27,33,39,53,67,69,75
[8]7,29,43,49,55,57,65,71
[8]7,8,52,53,66,67,68,69
[8]7,9,15,51,53,67,69,87
[8]7,9,27,33,53,67,69,87
[8]7,9,29,33,43,69,79,85
[8]7,9,29,43,49,69,71,85
[8]8,10,50,52,60,66,68,70
[8]8,11,49,52,57,66,68,71
[8]8,12,16,44,52,68,76,84
[8]8,12,26,34,52,66,68,86
[8]8,12,28,32,48,52,68,88
[8]8,12,28,34,44,50,80,86
[8]8,12,48,52,54,66,68,72
[8]8,13,47,51,52,66,68,73
[8]8,14,20,50,52,58,66,86
[8]8,14,28,48,50,64,74,80
[8]8,14,46,48,52,66,68,74
[8]8,15,21,27,52,66,68,87
[8]8,15,21,52,57,66,68,81
[8]8,15,25,35,52,66,68,85
[8]8,15,51,52,57,63,66,68
[8]8,16,42,44,52,66,68,76
[8]8,17,39,43,52,66,68,77
[8]8,18,28,32,36,52,68,88
[8]8,18,36,42,52,66,68,78
[8]8,19,33,41,52,66,68,79
[8]8,20,30,40,52,66,68,80
[8]8,21,23,37,52,66,68,83
[8]8,21,27,39,52,66,68,81
[8]8,27,33,39,52,66,68,75
[8]8,9,15,51,52,66,68,87
[8]8,9,27,33,52,66,68,87
[8]8,9,33,52,66,68,69,75
[8]8,9,51,52,63,66,68,69
[8]9,10,15,50,51,60,70,87
[8]9,10,27,33,50,60,70,87
[8]9,10,33,50,60,69,70,75
[8]9,10,50,51,60,63,69,70
[8]9,11,15,49,51,57,71,87
[8]9,11,17,23,25,49,83,89
[8]9,11,17,25,49,53,69,89
[8]9,11,23,37,49,71,75,83
[8]9,11,25,35,49,69,71,85
[8]9,11,27,29,31,49,71,89
[8]9,11,31,35,37,71,77,83
[8]9,11,31,35,67,69,71,77
[8]9,11,49,51,57,63,69,71
[8]9,12,15,24,48,51,84,87
[8]9,12,15,26,34,51,86,87
[8]9,12,15,48,51,54,72,87
[8]9,12,24,27,33,48,84,87
[8]9,12,24,33,48,69,75,84
[8]9,12,24,48,51,63,69,84
[8]9,12,26,27,33,34,86,87
[8]9,12,26,33,34,69,75,86
[8]9,12,26,34,51,63,69,86
[8]9,12,27,33,48,54,72,87
[8]9,12,33,48,54,69,72,75
[8]9,12,48,51,54,63,69,72
[8]9,13,23,29,49,51,55,89
[8]9,13,23,49,51,55,59,87
[8]9,13,27,33,47,51,73,87
[8]9,13,33,47,51,69,73,75
[8]9,14,15,46,48,51,74,87
[8]9,14,27,33,46,48,74,87
[8]9,14,33,46,48,69,74,75
[8]9,14,46,48,51,63,69,74
[8]9,15,16,42,44,51,76,87
[8]9,15,17,29,31,43,77,89
[8]9,15,18,24,36,51,84,87
[8]9,15,18,36,42,51,78,87
[8]9,15,19,33,41,51,79,87
[8]9,15,20,30,40,51,80,87
[8]9,15,21,23,37,51,83,87
[8]9,15,22,24,38,51,82,87
[8]9,15,25,27,33,35,85,87
[8]9,15,25,35,51,63,69,85
[8]9,16,27,33,42,44,76,87
[8]9,16,33,42,44,69,75,76
[8]9,16,42,44,51,63,69,76
[8]9,17,19,23,25,33,83,89
[8]9,17,19,25,33,53,69,89
[8]9,17,23,29,43,49,55,89
[8]9,17,23,37,43,63,77,83
[8]9,17,23,43,49,55,59,87
[8]9,17,27,33,39,43,77,87
[8]9,17,33,39,43,69,75,77
[8]9,18,24,27,33,36,84,87
[8]9,18,24,33,36,69,75,84
[8]9,18,24,36,51,63,69,84
[8]9,18,27,33,36,42,78,87
[8]9,18,33,36,42,69,75,78
[8]9,18,36,42,51,63,69,78
[8]9,19,33,41,51,63,69,79
[8]9,20,27,30,33,40,80,87
[8]9,20,30,33,40,69,75,80
[8]9,20,30,40,51,63,69,80
[8]9,21,23,27,33,37,83,87
[8]9,22,24,27,33,38,82,87
[8]9,22,24,33,38,69,75,82
[8]9,22,24,38,51,63,69,82
[8]9,23,25,33,35,37,83,85
[8]9,23,29,33,37,43,79,85
[8]9,23,29,37,43,49,71,85
[8]9,29,31,37,43,65,71,77
[8]9,29,31,37,51,65,71,73
[8]9,31,33,35,37,41,77,83
[8]9,31,33,35,41,67,69,77

northwolves · 发表于 2024-3-23 17:14:14

$C_44^2=946$: {a,b,90-b,90-a},0,a<b<45

mathe · 发表于 2024-3-23 18:10:29

计算结果好像还有

Special value 1 59 61 87 1.00000000000000
Special value 2 58 62 84 1.00000000000000
Special value 3 29 31 89 1.00000000000001
Special value 3 39 75 81 1.00000000000000
Special value 3 57 63 81 1.00000000000000
Special value 3 63 69 75 1.00000000000000
Special value 4 56 64 78 1.00000000000000
Special value 5 55 65 75 1.00000000000000
Special value 6 28 32 88 1.00000000000000
Special value 6 42 66 78 1.00000000000000
Special value 6 54 66 72 1.00000000000000
Special value 7 53 67 69 1.00000000000000
Special value 8 52 66 68 1.00000000000000
Special value 9 15 51 87 1.00000000000000
Special value 9 27 33 87 1.00000000000000
Special value 9 33 69 75 1.00000000000000
Special value 9 51 63 69 1.00000000000000
Special value 10 50 60 70 1.00000000000000
Special value 11 49 57 71 1.00000000000000
Special value 12 24 48 84 1.00000000000000
Special value 12 26 34 86 1.00000000000000
Special value 12 48 54 72 1.00000000000000
Special value 13 47 51 73 1.00000000000000
Special value 14 46 48 74 1.00000000000000
Special value 15 21 27 87 1.00000000000000
Special value 15 21 57 81 1.00000000000000
Special value 15 25 35 85 1.00000000000000
Special value 15 51 57 63 1.00000000000000
Special value 16 42 44 76 1.00000000000000
Special value 17 39 43 77 1.00000000000000
Special value 18 24 36 84 1.00000000000000
Special value 18 36 42 78 1.00000000000000
Special value 19 33 41 79 1.00000000000000
Special value 20 30 40 80 1.00000000000000
Special value 21 23 37 83 1.00000000000000
Special value 21 27 39 81 1.00000000000000
Special value 22 24 38 82 1.00000000000000
Special value 27 33 39 75 1.00000000000000

复制代码

王守恩 · 发表于 2024-3-23 20:41:52

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_ ...

38组解中的28组解(4,6,10,14,15,16,20,25,26,28不行)可以由恒等式推出来。$\frac{\tan(a)*\tan(\pi/3+a)\ \ }{\tan(3a)*\tan(\pi/6+a)\ \ }=1$ a好像可以是任意数。

northwolves · 发表于 2024-3-23 23:50:51

v=Select[Subsets[Range@89,{4}],N[Times@@Tan[# Degree]]==1&&#[[1]]+#[[4]]!=90&];{Length@v,v}

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{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},{5,55,65,75},{6,28,32,88},{6,42,66,78},{6,54,66,72},{7,53,67,69},{8,52,66,68},{9,15,51,87},{9,27,33,87},{9,33,69,75},{9,51,63,69},{10,50,60,70},{11,49,57,71},{12,24,48,84},{12,26,34,86},{12,48,54,72},{13,47,51,73},{14,46,48,74},{15,21,27,87},{15,21,57,81},{15,25,35,85},{15,51,57,63},{16,42,44,76},{17,39,43,77},{18,24,36,84},{18,36,42,78},{19,33,41,79},{20,30,40,80},{21,23,37,83},{21,27,39,81},{22,24,38,82},{27,33,39,75}}}

王守恩 · 发表于 2024-3-24 07:12:53

题目源自知乎《求证：tan15°=tan3°tan39°tan81°，这个三角函数问题怎么证明？》
也就是这里的第4道题：04--Special value 3 39 75 81 1.00000000000000
知乎作者作得辛苦, 知乎读者读得更辛苦。不知可有妙招,  把这38道题一并证明了。

又：连接网友 wayne帖子《[原创] 三角形的角格点问题》，里面有
四边形的4个角都是整数度数°,  四边形形内部存在一点P，
使得四边形的8个内角也都是整数度数°，每个四边形都可以有这样的P点。
我就特别好奇：这38道题都可以有这样的P点吗(四边形4个角都是90°;  P相关的4个角都是90°)？

mathe · 发表于 2024-3-24 10:27:15

这个题目类似三角形角格点问题，我们可以将整数度数扩展到有理度数。
由于题目等价于$\sin(a\degree)\sin(b\degree)\sin(c\degree)\sin(d\degree)-\cos(a\degree)\cos(b\degree)\cos(c\degree)\cos(d\degree)=0$
我们分别记$A=\exp(\frac{i a\pi}{180}),B=\exp(\frac{i b\pi}{180}),C=\exp(\frac{i c\pi}{180}),D=\exp(\frac{i d\pi}{180})$, 于是题目可以等价转化为
$(A-\frac1A)(B-\frac1B)(C-\frac1C)(D-\frac1D)=(A+\frac1A)(B+\frac1B)(C+\frac1C)(D+\frac1D)$
展开后得到
$A^2+B^2+C^2+D^2+B^2C^2D^2+A^2C^2D^2+A^2B^2D^2+A^2B^2C^2=0$
于是我们这个方程要求8个单位根之和为0。
根据链接关联的论文 https://math.mit.edu/~poonen/papers/ngon.pdf 中TABLE 1结论，
权重之和不超过8的单位根之和为0的本源组合只有

权重	模式	模式变种数目
2	$R_2$	1
3	$R_3$	1
5	$R_5$	1
6	$(R_5:R_3)$	1
7	$(R_5:2R_3)$	2
7	$R_7$	1
8	$(R_5:3R_3)$	2
8	$(R_7:R_3)$	1

由于不存在权重为1的方案，权重为7的不用考虑。所以我们只需要考虑权重分别为
8, 6+2, 5+3, 3+3+2, 2+2+2+2这5种组合

mathe · 发表于 2024-3-24 11:02:42

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$x+\omega_7,x+2\omega_7,x+3\omega_7,x+4\omega_7,x+5\omega_7,x+6\omega_7,x-\omega_3,x-2\omega_3$
其中$\omega_7,\omega_3$分别为$\frac{2\pi}7,\frac{2\pi}3$
而我们又要求这些角度分别为8个单位根$A^2,B^2,C^2,D^2,(BCD)^2,(ACD)^2,(ABD)^2,(ABC)^2$的幅角。
注意到8个单位根两两乘积相等，也就是要求这8个角度两两相加为等角。显然这8个角是无法达成这个目标的，所以需要淘汰。
ii)R5:3R3, 也就是$x,x+\omega_5,x+2\omega_5,x+3\omega_5,x+4\omega_5$中淘汰三个角度，拼接上$\omega_3$
由于表格已经给出本质上只有两种不等价组合，我们可以查看留下两个角度分别相邻和不相邻两种不同情况，对应
$x-3\omega_5,x-4\omega_5, x+\omega_3,x+2\omega_3, x+\omega_5+\omega_3,x+\omega_5+2\omega_3, x+2\omega_5+\omega_3,x+2\omega_5+2\omega_3$ 以及
$x-2\omega_5,x-4\omega_5, x+\omega_3,x+2\omega_3, x+\omega_5+\omega_3,x+\omega_5+2\omega_3, x+3\omega_5+\omega_3,x+3\omega_5+2\omega_3$。
其中第一种所有角度和为$8x-\omega_5+9\omega_3$,注意到$5\omega_5,3\omega_3$都是$2\pi$可以省略，所以8个角分成四组，每组内两个角的和都只能等于$2x-\frac{\omega_5}4 + h \frac{\pi}2$, 显然这个目标也无法达成，不符合要求。类似，第二种有不行。
所以权重为8的本源解都不符合本题要求。

现在我们再查看本源解为6+2类型的，其中权重6只能$R_5:R_3$,也就是6个角度可以为
$x+\omega_5,x+2\omega_5,x+3\omega_5,x+4\omega_5,x-\omega_3,x-2\omega_3$
而权重为2的为$R_2$,对应2两个角度$y,y+\omega_2$.
注意到前面6个角任意两个之间差值都不是$\pi$而后面两个差值为$\pi$,所以后面两个角如果不相互匹配，那么它们匹配对象差值应该为$\pi$,
得到矛盾，也就是后面两个角只能分到一组，也就是两个角度之和是$2y+\pi$。同时要求前面6个角分成3组，两两之和相等。
于是只能$x-\omega_3,x-2\omega_3$一组，$x+\omega_5,x+4\omega_5$一组，$x+2\omega_5,x+3\omega_5$,
而且每组内两个角度之和为$2x=2y+\pi$, 得到$y=x+\frac{\pi}2$
得到这8个角度为$x+\omega_5, x+4\omega_5; x+2\omega_5,x+3\omega_5; x-\omega_3,x-2\omega_3; x+\frac{\pi}2,x+\frac{3\pi}2$.
但是另外一方面，比如$A^2$幅角为$x+\omega_5$,我们会要求$B^2C^2D^2$幅角为$x+4\omega_5$, 也就是后面三组各条选择一个，要求和为$x+4\omega_5$等等，这样下去好像有会无解，但是计算量有些大，可能还是需要借助计算机分析，枚举各种不同的排列情况。

通过程序解放长可以发现有解的有
5+3 R5 R3:

Solution candiate with MOD 15:
2a = +1V1+0
2b = +1V1+3
2c = +1V1+6
2d = +1V2+0
Constrain:
+1V0-2V1+3 Mod 15=0
+1V1-1V2+1 Mod 15=0
+2V2+11 Mod 15=0
Solution candiate with MOD 15:
2a = +1V1+0
2b = +1V1+3
2c = +1V1+6
2d = +1V2+0
Constrain:
+1V0-2V1+3 Mod 15=0
+1V1-1V2+11 Mod 15=0
+2V2+1 Mod 15=0
Solution candiate with MOD 15:
2a = +1V1+0
2b = +1V1+3
2c = +1V1+6
2d = +1V2+0
Constrain:
+1V0-1V1-1V2+10 Mod 15=0
+1V1-1V2+10 Mod 15=0
+2V2+14 Mod 15=0
...

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3+3+2 R3 R3 R2

Solution candiate with MOD 6:
2a = +1V1+0
2b = +1V1+2
2c = +1V1+4
2d = +1V2+0
Constrain:
+1V0-1V1-1V2+4 Mod 6=0
+1V1-1V3+5 Mod 6=0
+1V2-1V3=0
+2V3=0
Solution candiate with MOD 6:
2a = +1V1+0
2b = +1V1+2
2c = +1V1+4
2d = +1V2+0
Constrain:
+1V0-1V1-1V2+4 Mod 6=0
+1V1-1V3+2 Mod 6=0
+1V2-1V3+3 Mod 6=0
+2V3=0
Solution candiate with MOD 6:
2a = +1V1+0
2b = +1V1+2
2c = +1V1+4
2d = +1V2+0
Constrain:
+1V0-1V1-1V2+2 Mod 6=0
+1V1-1V3+1 Mod 6=0
+1V2-1V3=0
+2V3=0
...

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2+2+2+2 R2 R2 R2 R2:

Solution candiate with MOD 2:
2a = +1V1+0
2b = +1V1+1
2c = +1V2+0
2d = +1V2+1
Constrain:
+1V0-1V1-1V3=0
+1V1-1V2+1V3-1V4=0
+3V2-2V3+1V4=0
Solution candiate with MOD 2:
2a = +1V1+0
2b = +1V1+1
2c = +1V2+0
2d = +1V2+1
Constrain:
+1V0-1V1-1V3=0
+1V1-1V2+1V3-1V4+1 Mod 2=0
+3V2-2V3+1V4+1 Mod 2=0
Solution candiate with MOD 2:
2a = +1V1+0
2b = +1V1+1
2c = +1V2+0
2d = +1V2+1
Constrain:
+1V0-1V1-1V3=0
+1V1-1V2+1 Mod 2=0
+3V2-1V3+1 Mod 2=0
+1V3-1V4+1 Mod 2=0
....

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现在代码给出解的方案还是太多了，主要原因在于有些等价的方案会被重复求解。
比如对于最后一种2+2+2+2, 我们会引入约束变量V1,V2,V3,V4等,这四个变量之间本质上是等价的，可以任意置换，得出看似不同的解，代码没有考虑。
另外V1和V1+1Mod2也可以相互置换等。类似3+3+2模式V1,V2可以置换,V3,V3+3可以置换，V1,V1+2,V1+4可以轮换，V2,V2+2,V2+4可以轮换等。
另外需要注意上面方程中常数项使用1Mod6之类的表示，代表将一个周角(360度)均分为6分，所以1Mod6实际上代表60度角。

比如我们解读第一种中第一个解:
Solution candiate with MOD 15:
2a = +1V1+0
2b = +1V1+3
2c = +1V1+6
2d = +1V2+0
Constrain:
+1V0-2V1+3 Mod 15=0
+1V1-1V2+1 Mod 15=0
+2V2+11 Mod 15=0
于是可以V2=2 Mod 15, V1=1 Mod 15, 代入得到2a=1 Mod 15, 2b=4 Mod 15, 2c=7 Mod 15, 2d = 2Mod15
对应$a=\frac{\pi}{15},b=\frac{4\pi}{15},c=\frac{7\pi}{15},d=\frac{2\pi}{15}$.
由此我们得到$\tan(\frac{\pi}{15})\tan(\frac{4\pi}{15})\tan(\frac{7\pi}{15})\tan(\frac{2\pi}{15})=1$
另外需要注意表达式中同余仅适用于常数项，非常数项是可以为分数的。准确的说，我们可以将上面表达式改为
2a = +1V1+0
2b = +1V1+3*2\pi/15
2c = +1V1+6*2\pi/15
2d = +1V2+0
Constrain:
+1V0-2V1+3*2\pi/15=整数*2\pi
+1V1-1V2+1*2\pi/15 =整数*2\pi
+2V2+11*2\pi/15=整数*2\pi
比如可以选择V2=$\frac{19\pi}{15}$,可以得到$\tan(\frac{17\pi}{30})\tan(\frac{23\pi}{30})\tan(\frac{29\pi}{30})\tan(\frac{19\pi}{30})=1$

进一步更新程序，将不含可变参数的方程全部解出，存放在附件中sv.e?u文件中，然后余下至少含有一个可变参数的放入sv.o?文件。

比如sv.o5中第一个方程对应恒等式$\tan(x-\frac{3\pi}6)\tan(x-\frac{\pi}6)\tan(x+\frac{\pi}6)\tan(3x)=1$

sv.tgz (15.3 KB, 下载次数: 4) (附件中是程序找到的候选解)

northwolves · 发表于 2024-3-24 11:07:10

$\tan (x) \tan \left(\frac{\pi }{3}-x\right) \tan \left(x+\frac{\pi }{3}\right)=\tan (3 x)$

s = Table[Sort@{x, 60 - x, 60 + x, 90 - 3 x}, {x, 29}]

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{{1,59,61,87},{2,58,62,84},{3,57,63,81},{4,56,64,78},{5,55,65,75},{6,54,66,72},{7,53,67,69},{8,52,66,68},{9,51,63,69},{10,50,60,70},{11,49,57,71},{12,48,54,72},{13,47,51,73},{14,46,48,74},{15,45,45,75},{16,42,44,76},{17,39,43,77},{18,36,42,78},{19,33,41,79},{20,30,40,80},{21,27,39,81},{22,24,38,82},{21,23,37,83},{18,24,36,84},{15,25,35,85},{12,26,34,86},{9,27,33,87},{6,28,32,88},{3,29,31,89}}

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[讨论] 这样的数组有多少？

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