901数系中的素数和倒数循环节长度

yangchuanju · 发表于 2025-10-25 06:42

901数系中的素数和倒数循环节长度

当n是正整数时10^n+1都是100…01型的整数，整数中0的个数等于n-1；（当n=1时，10^1+1=11中0的个数等于0）
当n是大于等于3的正奇数时10^n+1都是11的倍数数，(10^n+1)/11都是9090…91型的整数，整数中90的个数等于(n-1)/2；（当n=3时，(10^3+1)/11=91中90的个数等于0）
9090…9091型的整数全由9、0、1三个数字组成，在本贴中简称901数系。
901数系中的数字有素数，也有合数；素数较少，合数居多——
已知的素数有
1 9091
2 909091
3 909090909090909091
4 909090909090909090909090909091
5 9090909090909090909090909090909090909090909090909091
6 909090909090909090909090909090909090909090909090909090909090909091
（见A097209）

yangchuanju · 发表于 2025-10-25 06:44

A054416给出12个 9090...9091型素数的90+91的个数
Numbers k such that 9090...9091 (with k-1 copies of 90 and one copy of 91) is prime.
1 2——即9091
2 3——即909091
3 9
4 15
5 26
6 33
7 146
8 320
9 1068
10 1505
11 134103
12 800393

一定还有更多更大的9090...9091型素数，猜想有无穷多的！

yangchuanju · 发表于 2025-10-25 06:46

本帖最后由 yangchuanju 于 2025-10-30 03:45 编辑

（接上楼）
当90和91的总个数k等于1时91=7*13是合数；
k=2时9091是素数；
k=3时909091是素数；
k=4时90909091=7*13*19* 52579是合数；
k=5时9090909091=11*23*4093*8779是合数；
k=6时909090909091=859*1058313049<10>是合数；
k=7时90909090909091=7*13*211*241*2161*9091是合数；
k=8时9090909090909091=103*4013*21993833369<11>是合数；
k=9时909090909090909091是素数；
k=10时90909090909090909091=7^2*13*127*2689*459691* 909091是合数；
……

yangchuanju · 发表于 2025-10-25 06:52

本帖最后由 yangchuanju 于 2025-10-30 03:48 编辑

901数系倒数循环节长度
91及其素因子7,13的倒数循环节长度都是6；
积91的倒数循环节长度等于它的两个素因子倒数循环节长度的最小公倍数，6和6 的最小公倍数还是6；
素数9091的倒数循环节长度是10；
素数909091的倒数循环节长度是14；
k=4时90909091及其素因子19和52579的倒数循环节长度都是18；另两个素因子7和13的倒数循环节长度仍是6；
19*52579= 999001 也是901数系中的数
k=5时9090909091及其素因子23、4093和 8779的倒数循环节长度都是22；另一个素因子11的倒数循环节长度仍是2，但第2个素因子11的倒数循环节长度为22；
11*23*4093*8779= 9090909091 也是901数系中的数
k=6时909090909091及其素因子859和1058313049<10>的倒数循环节长度都是26；
k=7时90909090909091及其素因子211、241和2161的倒数循环节长度都是30；另两个素因子7和13的倒数循环节长度仍是6，9091的倒数循环节长度仍是10；
k=8时9090909090909091及其素因子103、4013和21993833369<11>的倒数循环节长度都是34；
k=9时素数909090909090909091倒数循环节长度是38；
k=10时90909090909090909091及其素因子127*2689*459691的倒数循环节长度都是42；素因子 909091的倒数循环节长度仍是14；素因子13的倒数循环节长度仍是6；另一个素因子7的倒数循环节长度仍是6，但第2个素因子7的倒数循环节长度为42；
……

yangchuanju · 发表于 2025-10-25 07:03

9901型素数
A187868
Primes of the form 100^k - 10^k + 1.
1 9901
2 99990001
3 999999000001
4 9999999900000001

Next term has k > 1000000. - Robert Gelhar, Aug 20 2020

yangchuanju · 发表于 2025-10-25 07:32

9个数分解式循环节长
1 91=7*13 6
2 9901 is prime 12
3 999001=19*52579 18
4 99990001 is prime 24
5 9999900001=7*13*211*241*2161 30
6 999999000001 is prime 36
7 99999990000001=7*7*13*127*2689*459691 42
8 9999999900000001 is prime 48
9 999999999000000001=70541929*14175966169 54
10 99999999990000000001=61*9901*4188901*39526741 60
11 9999999999900000000001=7*13*599144041*183411838171 66
12 999999999999000000000001=3169*98641*3199044596370769 72

yangchuanju · 发表于 2025-10-25 17:34

本帖最后由 yangchuanju 于 2025-10-25 17:55 编辑

更多个901型素数
根据A007618提供数据摘录

900900900900990990990991
4个900+3个990+1个991 24位

900900900900900900900900900900990990990990990990990990990991
10个900+9个990+1个991 60位

999999999999990000000000000099999999999999000000000000009999999999999900000000000001
2组(14个9+14个0)+(14个9+13个0+1个1） 84位

9999999999999999999999999999999900000000000000000000000000000000
9999999999999999999999999999999900000000000000000000000000000001
(32个9+32个0)+(32个9+31个0+1个1） 128位

yangchuanju · 发表于 2025-10-25 17:40

本帖最后由 yangchuanju 于 2025-10-26 05:53 编辑

几个大901型素数
9090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909091——292位，145个90+1个91，倒数循环节长度586

900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900900990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990990991——396位，66个900+65个990+1个991，倒数循环节长度597

99009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009901——584位，145个9900+1个9901，倒数循环节长度1172

9090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909090909091——640位，319个90+1个91，倒数循环节长度1282

yangchuanju · 发表于 2025-10-25 17:41

（接上楼）
999999999999999999999999999999999999999
000000000000000000000000000000000000000
000000000000000000000000000000000000000
999999999999999999999999999999999999999
000000000000000000000000000000000000000
000000000000000000000000000000000000000
999999999999999999999999999999999999999
000000000000000000000000000000000000000
000000000000000000000000000000000000000
999999999999999999999999999999999999999
000000000000000000000000000000000000000
000000000000000000000000000000000000000
999999999999999999999999999999999999999
999999999999999999999999999999999999999
000000000000000000000000000000000000000
999999999999999999999999999999999999999
999999999999999999999999999999999999999
000000000000000000000000000000000000000
999999999999999999999999999999999999999
999999999999999999999999999999999999999
000000000000000000000000000000000000000
999999999999999999999999999999999999999
999999999999999999999999999999999999999
000000000000000000000000000000000000001
——936位，4组(39个9+78个0)+3组(78个9+39个0) +1组(78个9+38个0+1个1)，倒数循环节长度1521

99009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009900990099009901——920位，229个9900+1个9901，倒数循环节长度1844

999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990000999900009999000099990001——1128位，140个99990000+1个99990001，倒数循环节长度2264

yangchuanju · 发表于 2025-10-25 17:49

本帖最后由 yangchuanju 于 2025-10-25 18:04 编辑

几个较少的901型素数的构成及倒数循环节

9091 1个90+1个91，循环节0001099989，10位；
等于(10^5+1)/11，4位

9901 1个99+1个01，循环节000100999899，12位；
等于(10^12-1)/9/(3 · 7 · 11 · 13 · 37 · 101)，4位

909091 2个90+1个91，循环节00000109999989，14位；
等于(10^7+1)/11，6位

99990001 4个9+3个0+1个1，循环节000000010000999999989999，24位
等于(10^24-1)/9/(3 · 7 · 11 · 13 · 37 · 73 · 101 · 137 · 9901)，8位

999999000001 6个9+5个0+1个1，
循环节000000000001000000999999999998999999，36位；
等于(10^36-1)/9/(3^2 · 7 · 11 · 13 · 19 · 37 · 101 · 9901 · 52579 · 333667)，12位

909090909090909091 8个90+1个91，
循环节00000000000000000109999999999999999989，38位；
等于(10^19+1)/11，18位

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