233猜想

毒酒滴冻鸭

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。。。）

northwolves

1. f233[n_] :=NestWhileList[ With[{p = FactorInteger[#][[-1, 1]]}, 10 p + Mod[p, 10]] &, n, # != 233 &]; f233[477]

复制代码

随机测试了10^20以内的10万个数字，都返回233

mathe

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？

mathe

这个还可以试验一下八进制和十六进制结果

northwolves

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：

$f233(30k+1)=f233(30k+2)=f233(15k+1)=233$
$f233(30k+7)=f233(30k+14)=f233(15k+7)=233$
$f233(30k+11)=f233(30k+12)=f233(5k+2)=233$
$f233(30k+13)=f233(30k+16)=f233(15k+8)=233$
$f233(30k+17)=f233(30k+24)=f233(5k+4)=233$
$f233(30k+19)=f233(30k+28)=f233(15k+14)=233$
$f233(30k+23)=f233(30k+26)=f233(15k+13)=233$
$f233(30k+29)=f233(30k+38)=f233(15k+19)=233$

wayne

简单用Mathematica跑了下, n位数对应的最长环路的长度,逢177退出,

1. {59, 27}
   
2. {499, 34}
   
3. {4339, 45}
   
4. {14321, 51}
   
5. {807607, 55}
   
6. {1255609, 57}
   
7. {23244413, 61}

复制代码

wayne

9位数最长路径长度的输出如下,177退出.

1. 429872893,61
   
2. 518350409,61
   
3. 532297057,61
   
4. 859745786,61
   
5. 116222065,60
   
6. 139466478,60
   
7. 162710891,60
   
8. 185955304,60
   
9. 209199717,60
   
10. 232444130,60
    
11. 255688543,60
    
12. 278932956,60
    
13. 302177369,60
    
14. 325421782,60
    
15. 331624739,60
    
16. 348666195,60
    
17. 371910608,60
    
18. 395155021,60
    
19. 418399434,60
    
20. 441643847,60
    
21. 464888260,60
    
22. 488132673,60
    
23. 511377086,60
    
24. 534621499,60
    
25. 557865912,60
    
26. 581110325,60
    
27. 604354738,60
    
28. 627599151,60
    
29. 650843564,60
    
30. 663249478,60
    
31. 674087977,60
    
32. 697332390,60
    
33. 713294501,60
    
34. 720576803,60
    
35. 721117349,60
    
36. 743821216,60
    
37. 767065629,60
    
38. 790310042,60
    
39. 813554455,60
    
40. 824549851,60
    
41. 836798868,60
    
42. 860043281,60
    
43. 883287694,60
    
44. 906532107,60
    
45. 929776520,60
    
46. 944758273,60
    
47. 953020933,60
    
48. 976265346,60
    
49. 994874217,60
    
50. 999509759,60

复制代码

10位数正在运行中
.

wayne

10位数最长路径长度的输出如下,177退出.

1. 7722991429,64
   
2. 9639009451,64
   
3. 5073162869,63
   
4. 5504755537,63
   
5. 6468299053,63
   
6. 7200933221,63
   
7. 2301035707,62
   
8. 2670085721,62
   
9. 4305562619,62
   
10. 4602071414,62
    
11. 5340171442,62
    
12. 5867430859,62
    
13. 5975906137,62
    
14. 6136708639,62
    
15. 6248221373,62
    
16. 6903107121,62
    
17. 7703895799,62
    
18. 7845998617,62
    
19. 7993986073,62
    
20. 8010257163,62
    
21. 8441962499,62
    
22. 8611125238,62
    
23. 8804482717,62
    
24. 8995820263,62
    
25. 9204142828,62
    
26. 9240427219,62
    
27. 9758114671,62

复制代码

wayne

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.

wayne

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

1. {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}

复制代码

1. {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}

复制代码

f233函数见楼上的, 我加了个最多迭代100次.

1. f233[n_] := NestWhileList[With[{p = FactorInteger[#][[-1, 1]]}, 10 p + Mod[p, 10]] &, n, # != 233 &, 1, 100]

复制代码

1. t = 20; Monitor[
   
2. Do[p = RandomPrime[{10^30, 10^60}]; pp = f233[p]; tmp = Length[pp];
   
3.   If[tmp > t, Print[{tmp, pp}]; t = tmp], {m, 100000}], {m, t,
   
4.   Length[pp], p}]

复制代码