233猜想
在百度数学吧看到一个挺有趣的猜想，姑且直观名为【233猜想】：如图，对于任何大于1的正整数，取其最大素因数（如其本身为素数则取其本身），再在其后缀上这个素因数的尾数（如此新生成的整数最后两位必然重复）。。。
不断重复这过程，直至进入循环。。。
猜想：无论从任何大于1的正整数开始，最后必然进入上图这个26个正整数的循环。。。
（这个循环的最小正整数其实是177，但是233在华人网络圈子里比较有梗，所以索性用233好了。。。而且233这个位置有两个特性，首先是唯一开启连续三个素数的，另外它的前一项是唯一有四种不同素因数的。。。） f233 :=NestWhileList[ With[{p = FactorInteger[#][[-1, 1]]}, 10 p + Mod] &, n, # != 233 &]; f233
随机测试了10^20以内的10万个数字，都返回233 这个看上去应该问题不大，现在计算机已经验证了多大范围？ 这个还可以试验一下八进制和十六进制结果 一个不太严谨的证明：
对于素数$p$,若$f233(p)=true$, 则对于$q<=p,f233(q*p)=f233(p)=true$.
显然$f(233(p_1^{r_1}*p_2^{r_2}*p_3^{r_3}*...*p_k^{r_k})=f233(p_k)$
所以只判断素数即可
现在假设对于$2^m $以内的数字$n$,$f233(n)=233$
,对于$2^m-2^{m+1}$范围内的素数按模30分类：
$f233(30k+1)=f233(30k+2)=f233(15k+1)=233$
$f233(30k+7)=f233(30k+14)=f233(15k+7)=233$
$f233(30k+11)=f233(30k+12)=f233(5k+2)=233$
$f233(30k+13)=f233(30k+16)=f233(15k+8)=233$
$f233(30k+17)=f233(30k+24)=f233(5k+4)=233$
$f233(30k+19)=f233(30k+28)=f233(15k+14)=233$
$f233(30k+23)=f233(30k+26)=f233(15k+13)=233$
$f233(30k+29)=f233(30k+38)=f233(15k+19)=233$ 简单用Mathematica跑了下, n位数对应的最长环路的长度,逢177退出,
{59, 27}
{499, 34}
{4339, 45}
{14321, 51}
{807607, 55}
{1255609, 57}
{23244413, 61} 9位数最长路径长度的输出如下,177退出.
429872893,61
518350409,61
532297057,61
859745786,61
116222065,60
139466478,60
162710891,60
185955304,60
209199717,60
232444130,60
255688543,60
278932956,60
302177369,60
325421782,60
331624739,60
348666195,60
371910608,60
395155021,60
418399434,60
441643847,60
464888260,60
488132673,60
511377086,60
534621499,60
557865912,60
581110325,60
604354738,60
627599151,60
650843564,60
663249478,60
674087977,60
697332390,60
713294501,60
720576803,60
721117349,60
743821216,60
767065629,60
790310042,60
813554455,60
824549851,60
836798868,60
860043281,60
883287694,60
906532107,60
929776520,60
944758273,60
953020933,60
976265346,60
994874217,60
999509759,60
10位数正在运行中
. 10位数最长路径长度的输出如下,177退出.
7722991429,64
9639009451,64
5073162869,63
5504755537,63
6468299053,63
7200933221,63
2301035707,62
2670085721,62
4305562619,62
4602071414,62
5340171442,62
5867430859,62
5975906137,62
6136708639,62
6248221373,62
6903107121,62
7703895799,62
7845998617,62
7993986073,62
8010257163,62
8441962499,62
8611125238,62
8804482717,62
8995820263,62
9204142828,62
9240427219,62
9758114671,62 两个问题:
1) 收敛性. 这个迭代过程是否一定收敛,永不发散.
2) 唯一性. 如果收敛,是否一定只有一个固定的圈,而不是多个可能的圈. 随机搜索20-60位数, 发现了一个比较长的链条,71个数到达233
{1404959316591232018761438170532482914649639054831389666266107,14049593165912320187614381705324829146496390548313896662661077,10131211003848100461428493672419766391719761825466977,11170656435832081311382901699,2376735411879166236464447177,729664265458866618507499,2671784201606981393299,20036943861330311,34140303052199,341403030521999,463673815733,8864699,143211,477377,280811,2808111,550611,203933,43399,433999,100933,144199,131099,42299,422999,32299,322999,3229999,50399,4999,49999,499999,1277,12777,42599,10399,103999,1799,2577,8599,85999,859999,175511,250733,433,4333,6199,61999,5211,1933,19333,193333,3899,5577,133,199,1999,19999,28577,411,1377,177,599,5999,8577,9533,95333,136199,194577,8211,233}
{89268374898510845387137458290865198435238410091387,892683748985108453871374582908651984352384100913877,1062651827027926995872967741565622812982830133,200500344722250376579805234257664681694873611,1060591110334579148177,10605911103345791481777,557733242007866577,8772955199,87729551999,877295519999,22526933,143211,477377,280811,2808111,550611,203933,43399,433999,100933,144199,131099,42299,422999,32299,322999,3229999,50399,4999,49999,499999,1277,12777,42599,10399,103999,1799,2577,8599,85999,859999,175511,250733,433,4333,6199,61999,5211,1933,19333,193333,3899,5577,133,199,1999,19999,28577,411,1377,177,599,5999,8577,9533,95333,136199,194577,8211,233}
f233函数见楼上的, 我加了个最多迭代100次.
f233 := NestWhileList[[-1, 1]]}, 10 p + Mod] &, n, # != 233 &, 1, 100]
t = 20; Monitor[
Do; pp = f233; tmp = Length;
If; t = tmp], {m, 100000}], {m, t,
Length, p}]
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