求 n 使得 n! 以 2021 开头
某书证明了存在 n，使得 n! 可以以任意一串数字作为开头。现求正整数 n，使得 n! 以2021 开头。 1697917791
24764
26619
27727
34990
42004
57216
71869
74494
77981
81253
87457
98508 使用 Mathematica 硬算，竟然也好使：
n = 11; prod = 10!;
While[n < 100000, prod = prod*n;
If[], 4] == {2, 0, 2, 1}, Print];
n = n + 1
]
{16979,17791,24764,26619,27727,34990,42004,57216,71869,74494,77981,81253,87457,98508,101646,106752,115082,117536,123759,124963,125097,128930,129496,143176,145430,145608,147954,150084,159119,160231,161163,168079,170851,175620,179179,186731,194395,197814,200531,202408,205634,220996,229949,232031,232165,232359,233538,233812,236569,239429,241831,244379,256844,258008,260086,266522,274849,279710,282887,283345,283429,287093,297038,313125,320693,324548,327849,339662,343693,344403,347492,347553,348061,355546,357943,358607,359862,366076,369742,375529,384520,385526,411111,414986,427070,428915,442399,446228,446642,446722,452782,453888,456909,459849,463837,465062,465878,470988,474609,478127,480111,490862,496341,499823,502028,506166,507946,510457,517036,528199,532074,543024,551055,554680,555384,556545,562137,562545,573616,574616,591316,597692,603304,606644,608358,615043,616439,621847,624892,630419,631890,641452,654517,661103,661844,665506,667020,675154,676942,678469,679987,681658,682597,682784,687780,687940,689781,699900,708254,709378,710487,722016,727589,728845,732825,736823,743496,749697,754686,765342,766274,770955,777918,783395,785148,786380,793419,797545,797657,799510,807757,819293,843222,845303,847777,849418,858218,867231,872432,878030,888901,889824,894485,906501,908758,908999,913891,916410,916542,920553,922326,922440,930970,937076,946271,949542,953722,972282,972778,981548,982177,988996} 无聊的跑了10分钟，刷出了10^8的所有结果，突破了我就给代码，:lol
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单核版本的代码：
n=1;Monitor;If<t-(Floor]-3)Log<Log,Print]];n++]],n]
多核版本的代码：
CloseKernels[];LaunchKernels;
Timing<Last[#]-(Floor/Log]-3)Log<Log&]&,Table[{n,LogGamma},{n,10^8}]][];] 本帖最后由 uk702 于 2021-1-27 23:08 编辑
wayne 发表于 2021-1-27 21:46
单核版本的代码：
多核版本的代码：
我将你的多核版本的结果打印出来，似乎各有各的不准。
31419078!=2022000269906979489591697...，你的多了这个。
39600422!=2021000126106549851567511...，你的少了这个。
48400100!=2020999793431651954566836...，我的多了这个。
48844860!=2021000416575056021475393...，你的少了这个。
80495570!=2020999980052118200212907...，我的多了这个。
我实际给出的解是 21439 个（显示时多算了一个），多了 2 个，你给出了 21436 个，少了 2 个并多了 1 个。故检查后共同的解是 21437 个，暂不排除其中有我们都算错的。 这是有可能的，因为在做比较的时候用的是double精度,我稍微做了修改，用30位高精度计算，重新跑了下程序，速度慢了不少，算出正确的结果应该有21439 个。
CloseKernels[];LaunchKernels;
Timing<Last[#]-(Floor/Log]-3)Log<Log&]&,Table[{n,LogGamma},{n,10^8}]][];]发现前面的double精度的版本产生了错误解31419078，60459888，还少了39600422，48844860，61543437，88289946，90711389。 我的单核版本，julia 代码，double 精度，使用的某位同学论文中的斯特林加强公式（https://wenku.baidu.com/view/77419c44336c1eb91a375d3d.html），直接计算 log(n) 的累加和在 double 精度下累积误差似乎相当大。
t1=time();
h(n) = (log(sqrt(2.0*pi*n)) + n*log(n)) - n * (1-1/(0.4+12*n^2));
n=10; ct=1; u = log(2021.); v = log(2022.)
while n < 10^8
global n, ct
a = (h(n) - u)/log(10.); t = floor(Int, a);
if h(n) < v + t*log(10) ct = ct+1;println(n) end
n=n+1
end
# 打印出来的 ct 比实际多 1 个
println("find $ct sols, use time = $(time() - t1)")
斯泰林公式可以采用
\(\log(n!) \approx \frac{\log(2\pi n)}2+n\log(n)-n+\frac1{12n}+O(\frac{1}{n^3})\)
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