π和e的等位序列

数论爱好者

在1万位内，验证或者推翻下面这张图片的猜想：平均每隔10位数，π值和e值会有一个数字相同。
记第 n 个相同位为a(n)——称为π和e的等位序列, 记向前差分Δa(n)=a(n+1)-a(n), 猜想是说 $lim_{n→∞}E(Δa)≤10$

以下是比较前100位得到的等位序列的一个前段：
a(n)={13, 17, 18, 21, 34, 40, 45, 56, 59, 70, 81, 95, 100}
Δa(n)={4, 1, 3, 13, 6, 5, 11, 3, 11, 11, 14, 5}
我的猜想1: Δa(n)可以无穷大，并不是每隔10个左右就相同一次。
谁能率先找到Δa(n)=30的项。

猜想2：相同的数字中，9的出现率最高，或者能达到50%？

northwolves

A052055 Positions in both Pi and e indicate a common digit.

13, 17, 18, 21, 34, 40, 45, 56, 59, 70, 81, 95, 100, 143, 170, 206, 244, 263, 275, 279, 294, 324, 326, 331, 334, 361, 365, 388, 389, 396, 412, 420, 428, 429, 453, 460, 461, 462, 484, 494, 500, 501, 504, 507, 512, 523, 526, 548, 582, 591, 595, 596, 599, 603...

northwolves

294 <--> 324

Δa(n)={4, 1, 3, 13, 6, 5, 11, 3, 11, 11, 14, 5, 43, 27, 36, 38, 19, 12, 4, 15, 30, 2, 5, 3, 27, 4, 23, 1, 7, 16, 8, 8, 1, 24, 7, 1, 1, 22, 10, 6, 1, 3, 3, 5, 11, 3, 22, 34, 9, 4, 1, 3, 4,...}

数论爱好者

northwolves 发表于 2025-3-5 07:10
294  324

1000位中找到有100次左右相同的数字，9这个数字出现14次
10万位中找到有9998次相同数字，9这个数字出现了100次
根据以上数据：相同数字0到9的出现频率大致相等，都是在10%左右。
应该不会有其它新猜想成立，比如相同的数字出现在斐波那契数列数位上，等等，这些都不成立。它们都是无理数，无理可讲。

数论爱好者

数论爱好者 发表于 2025-3-5 08:23
1000位中找到有100次左右相同的数字，9这个数字出现14次
10万位中找到有9998次相同数字，9这个数字出现了 ...

10万位中，最大间隔87，85557-85470=87，出现1次。
大于50以上的比例比较少，总的不到60次。
间隔为1的出现比例很高，1038次，间隔为2的出现926次，以此逐渐减少下去

数论爱好者

数论爱好者 发表于 2025-3-5 09:02
10万位中，最大间隔87，85557-85470=87，出现1次。
大于50以上的比例比较少，总的不到60次。
间隔为1的出 ...

据此推断，可以找到两位数相同的数字，即Δa(n)=1的项。找不到3位，4位...更多位相同的数字，即Δa(n)中没有连续出现的 1。

数论爱好者

在 π 中可以找到你身份证号片段，在 e 中也可能找到你的身份证号片段。但在π和e中，这种18位片段的位置不可能相同，绝对不可能。

northwolves

数论爱好者 发表于 2025-3-5 09:12
据此推断，可以找到两位数相同的数字，即Δa(n)=1的项。找不到3位，4位...更多位相同的数字，即Δa(n)中没 ...

太武断了，理论上任意长的相同的数字都可能出现，如：

1. a=RealDigits[Pi,10,1000003][[1]];
   
2. b=RealDigits[E,10,1000003][[1]];k=3;x3=Select[Range@1000000,Take[a,{#-1,#+k-1}]==Take[b,{#-1,#+k-1}]&]
   
3. k=4;x4=Select[Range@1000000,Take[a,{#-1,#+k-1}]==Take[b,{#-1,#+k-1}]&]

复制代码

相同四位数出现位置：
{6425,19384,26550,26551,30744,33771,37239,60139,69950,72686,78655,84394,84395,86426,92288,108427,108908,127811,133746,146382,155331,188780,218715,224016,251862,265364,269785,269786,286666,300193,303003,303004,311266,323592,332811,332849,335798,336465,345021,363996,363997,372924,373485,375068,376583,381493,389644,437703,438707,450865,455440,456271,475908,482851,483485,489929,495006,504908,520741,529764,535783,538008,546634,552108,559882,590782,595451,595452,604727,604728,608641,632360,637457,637458,671950,672895,672896,676662,694514,698157,730781,744841,749857,753067,755659,765477,769034,769035,819140,868539,870409,871066,890296,892013,897401,918723,939811,953918,955239,960790,983944,983945,984067,987482,995416}

相同五位数出现位置：
{26550,84394,269785,303003,363996,595451,604727,637457,672895,769034,983944}

northwolves

五位数：{{26550,{8,2,0,1,4},{8,2,0,1,4}},{84394,{3,9,7,9,7},{3,9,7,9,7}},{269785,{0,0,0,9,3},{0,0,0,9,3}},{303003,{5,4,2,6,5},{5,4,2,6,5}},{363996,{3,6,6,6,6},{3,6,6,6,6}},{595451,{8,4,1,7,8},{8,4,1,7,8}},{604727,{2,4,4,9,9},{2,4,4,9,9}},{637457,{0,2,0,2,1},{0,2,0,2,1}},{672895,{3,7,2,9,1},{3,7,2,9,1}},{769034,{2,2,2,1,0},{2,2,2,1,0}},{983944,{9,5,5,9,3},{9,5,5,9,3}}}

northwolves

四位数：
{{6425,{0,2,1,3},{0,2,1,3}},{19384,{4,8,3,3},{4,8,3,3}},{26550,{8,2,0,1},{8,2,0,1}},{26551,{2,0,1,4},{2,0,1,4}},{30744,{5,1,8,4},{5,1,8,4}},{33771,{1,2,8,0},{1,2,8,0}},{37239,{8,2,5,7},{8,2,5,7}},{60139,{4,3,3,6},{4,3,3,6}},{69950,{5,1,3,2},{5,1,3,2}},{72686,{0,2,2,9},{0,2,2,9}},{78655,{9,6,5,2},{9,6,5,2}},{84394,{3,9,7,9},{3,9,7,9}},{84395,{9,7,9,7},{9,7,9,7}},{86426,{5,1,1,6},{5,1,1,6}},{92288,{0,7,3,6},{0,7,3,6}},{108427,{2,8,4,5},{2,8,4,5}},{108908,{7,6,2,2},{7,6,2,2}},{127811,{0,4,1,7},{0,4,1,7}},{133746,{2,9,0,1},{2,9,0,1}},{146382,{8,3,0,7},{8,3,0,7}},{155331,{3,7,1,3},{3,7,1,3}},{188780,{9,5,5,6},{9,5,5,6}},{218715,{8,1,9,7},{8,1,9,7}},{224016,{0,4,1,2},{0,4,1,2}},{251862,{0,2,9,7},{0,2,9,7}},{265364,{4,0,4,6},{4,0,4,6}},{269785,{0,0,0,9},{0,0,0,9}},{269786,{0,0,9,3},{0,0,9,3}},{286666,{1,3,2,5},{1,3,2,5}},{300193,{0,5,4,3},{0,5,4,3}},{303003,{5,4,2,6},{5,4,2,6}},{303004,{4,2,6,5},{4,2,6,5}},{311266,{1,5,3,4},{1,5,3,4}},{323592,{9,8,1,1},{9,8,1,1}},{332811,{3,2,9,7},{3,2,9,7}},{332849,{3,2,2,6},{3,2,2,6}},{335798,{3,2,5,8},{3,2,5,8}},{336465,{6,3,2,0},{6,3,2,0}},{345021,{3,5,9,1},{3,5,9,1}},{363996,{3,6,6,6},{3,6,6,6}},{363997,{6,6,6,6},{6,6,6,6}},{372924,{3,1,6,2},{3,1,6,2}},{373485,{5,5,3,1},{5,5,3,1}},{375068,{2,0,6,9},{2,0,6,9}},{376583,{8,1,8,6},{8,1,8,6}},{381493,{5,1,2,9},{5,1,2,9}},{389644,{5,2,5,1},{5,2,5,1}},{437703,{9,5,3,0},{9,5,3,0}},{438707,{6,0,8,3},{6,0,8,3}},{450865,{2,6,1,9},{2,6,1,9}},{455440,{8,1,7,0},{8,1,7,0}},{456271,{7,6,4,7},{7,6,4,7}},{475908,{6,5,9,5},{6,5,9,5}},{482851,{1,8,8,0},{1,8,8,0}},{483485,{7,5,9,1},{7,5,9,1}},{489929,{0,6,5,8},{0,6,5,8}},{495006,{9,7,7,5},{9,7,7,5}},{504908,{1,8,8,7},{1,8,8,7}},{520741,{5,6,8,1},{5,6,8,1}},{529764,{9,5,4,9},{9,5,4,9}},{535783,{4,8,8,3},{4,8,8,3}},{538008,{4,0,5,3},{4,0,5,3}},{546634,{2,9,7,3},{2,9,7,3}},{552108,{7,0,4,9},{7,0,4,9}},{559882,{1,6,9,5},{1,6,9,5}},{590782,{0,9,2,4},{0,9,2,4}},{595451,{8,4,1,7},{8,4,1,7}},{595452,{4,1,7,8},{4,1,7,8}},{604727,{2,4,4,9},{2,4,4,9}},{604728,{4,4,9,9},{4,4,9,9}},{608641,{6,4,4,6},{6,4,4,6}},{632360,{8,5,7,3},{8,5,7,3}},{637457,{0,2,0,2},{0,2,0,2}},{637458,{2,0,2,1},{2,0,2,1}},{671950,{5,5,5,2},{5,5,5,2}},{672895,{3,7,2,9},{3,7,2,9}},{672896,{7,2,9,1},{7,2,9,1}},{676662,{9,2,7,6},{9,2,7,6}},{694514,{7,8,8,0},{7,8,8,0}},{698157,{9,8,3,0},{9,8,3,0}},{730781,{0,7,8,9},{0,7,8,9}},{744841,{1,7,2,4},{1,7,2,4}},{749857,{4,6,1,2},{4,6,1,2}},{753067,{3,8,8,5},{3,8,8,5}},{755659,{6,2,0,4},{6,2,0,4}},{765477,{2,7,0,2},{2,7,0,2}},{769034,{2,2,2,1},{2,2,2,1}},{769035,{2,2,1,0},{2,2,1,0}},{819140,{7,4,3,2},{7,4,3,2}},{868539,{0,9,4,1},{0,9,4,1}},{870409,{7,7,0,4},{7,7,0,4}},{871066,{1,2,7,3},{1,2,7,3}},{890296,{7,6,4,6},{7,6,4,6}},{892013,{8,1,0,8},{8,1,0,8}},{897401,{2,9,2,4},{2,9,2,4}},{918723,{7,4,7,8},{7,4,7,8}},{939811,{3,0,9,4},{3,0,9,4}},{953918,{8,1,9,1},{8,1,9,1}},{955239,{8,4,4,4},{8,4,4,4}},{960790,{0,1,3,2},{0,1,3,2}},{983944,{9,5,5,9},{9,5,5,9}},{983945,{5,5,9,3},{5,5,9,3}},{984067,{2,0,8,1},{2,0,8,1}},{987482,{0,9,7,5},{0,9,7,5}},{995416,{0,3,3,3},{0,3,3,3}}}