求证:最大素数各位数字辗转式相加个位数是1

太阳 · 发表于 2021-2-25 05:51

已知:整数a>0，素数e，f，m，...，k
333667333667333667...333667/3=a，a=e*f*m*...*k
求证:最大素数各位数字辗转式相加个位数是1

yangchuanju · 发表于 2021-2-25 11:06

依题意，合数a只能是：
3 333667333667333667
6 333667333667333667333667333667333667
9 333667333667333667333667333667333667333667333667333667
12 333667333667333667333667333667333667333667333667333667333667333667333667
15 333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667
18 333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667
21 333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667
24 333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667
27 333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667
30 333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667
分解之并计算各最大素数辗转相加结果
=================================
3 333667333667333667=P1 * P2 * P5 * P6 * P6
P1 = 3
P2 = 19
P5 = 52579
P6 = 333667
P6 = 333667 3+3+3+6+6+7=28,2+8=10,1+0=1
cputime 0:00:00:00
=================================
6 333667333667333667333667333667333667=P1 * P2 * P3 * P4 * P5 * P6 * P6 * P12
P1 = 3
P2 = 19
P3 = 101
P4 = 9901
P5 = 52579
P6 = 333667
P6 = 333667
P12 = 999999000001 9*6+1=55,5+5=10,1+0=1
cputime 0:00:00:03
=================================
9 333667333667333667333667333667333667333667333667333667=P1 * P1 * P2 * P3 * P5 * P6 * P6 * P8 * P11 * P15
P1 = 3
P1 = 3
P2 = 19
P3 = 757
P5 = 52579
P6 = 333667
P6 = 333667
P8 = 70541929 7+5+4+1+9+2+9=37,3+7=10,1+0=1
P11 = 14175966169 1+4+1+7+5+9+6+6+1+6+9=55，5+5=10,1+0=1
P15 = 440334654777631 4+4+3+3+4+6+5+4+7+7+7+6+3+1=64,6+4=10,1+0=1
cputime 0:00:00:15
=================================
12 333667333667333667333667333667333667333667333667333667333667333667333667=P1 * P2 * P2 * P3 * P3 * P4 * P4 * P5 * P5 * P6 * P6 * P8 * P12 * P16
P1 = 3
P2 = 19
P2 = 73
P3 = 101
P3 = 137
P4 = 3169
P4 = 9901
P5 = 52579
P5 = 98641
P6 = 333667
P6 = 333667
P8 = 99990001 9*4+1=37,3+7=10,1+0=1
P12 = 999999000001 9*6+1=55,5+5=10,1+0=1
P16 = 3199044596370769 3+1+9+9+4+4+5+9+6+3+7+7+6+9=82,8+2=10,1+0=1
cputime 0:00:00:17
=================================
15 333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667=P1 * P2 * P2 * P2 * P3 * P3 * P3 * P4 * P4 * P5 * P5 * P6 * P6 * P6 * P7 * P7 * P13 * P19
P1 = 3
P2 = 19
P2 = 31
P2 = 41
P3 = 211
P3 = 241
P3 = 271
P4 = 2161
P4 = 9091
P5 = 29611
P5 = 52579
P6 = 238681 2+3+8+6+8+1=28,2+8=10,1+0=1
P6 = 333667
P6 = 333667
P7 = 2906161 2+9+6+1+6+1=25，2+5=7
P7 = 3762091 3+7+6+2+9+1=28,2+8=10,1+0=1
P13 = 8985695684401 8+9+8+5+6+9+5+6+8+4+4+1=73,7+3=10,1+0=1
P19 = 4185502830133110721 4+1+8+5+5+2+8+3+1+3+3+1+1+7+2+1=55,5+5=10,1+0=1
cputime 0:00:01:95
=================================
18 333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667333667=P1 * P1 * P2 * P3 * P3 * P3 * P4 * P5 * P6 * P6 * P6 * P8 * P11 * P12 * C44
P1 = 3
P1 = 3
P2 = 19
P3 = 101
P3 = 109
P3 = 757
P4 = 9901
P5 = 52579
P6 = 153469 1+5+3+4+6+9=28,2+8=10,1+0=1
P6 = 333667
P6 = 333667
P8 = 70541929 7+5+4+1+2+9=28,2+8=10,1+0=1
P11 = 14175966169 1+4+1+7+5+9+6+6+1+6+9=55,5+5=10,1+0=1
P12 = 999999000001 9*6+1=55,5+5=10,1+0=1
C44 = 26323019469887323242082320257150758321292311
cputime 0:00:04:79
=================================
这种素数纯属巧合，不可能进行数学证明，无学术价值，只起消闲作用！

yangchuanju · 发表于 2021-2-25 11:33

本帖最后由 yangchuanju 于 2021-2-25 11:35 编辑

10^10-1含一个4位素因子9091
10^14-1含一个6位素因子909091
10^38-1含一个18位素因子909090909090909091
10^62-1含一个30位素因子909090909090909090909090909091
10^106-1含一个52位素因子9090909090909090909090909090909090909090909090909091
10^134-1含一个66位素因子909090909090909090909090909090909090909090909090909090909090909091

10的18,22,26,30,34次方减1应各含一个8,10,12,14,16位合数因子，请有兴趣的老师或网友列出对应数字并分解！
下一个同类素因子的指数是586和1282，它们各多少位？

yangchuanju · 发表于 2021-2-25 13:23

本帖最后由 yangchuanju 于 2021-2-25 13:25 编辑

再来一个同类型的题目，
10^12-1含一个4位素因子9901
10^1172-1含一个584位素因子99009900…9901
10^1844-1含一个920位素因子99009900…9901
依照比例关系，10^20-1中应有一个8位的因子99009901，
现知10^20-1的所有素因子是：
3 3 11 41 101
271 3541 9091 27961
经试算3541*27961=99009901
又知10^28-1应有一个12位的因子990099009901，
现知10^28-1的所有素因子是：
3 3 11 29 101
239 281 4649 909091 121499449
经试算29*281*121499449=990099009901
预测正确。

请分解10^36-1并找出哪些素因子的积等于9900990099009901？

yangchuanju · 发表于 2021-2-25 13:28

10^12-1含一个4位素因子9901
10^24-1含一个8位素因子99990001
10^36-1含一个12位素因子999999000001
10^48-1含一个16位素因子9999999900000001
指数再大就没有这种类型的素因子了。
请分解10^60-1并找出哪些素因子的积等于99999999990000000001（20位）？
请分解10^72-1并找出哪些素因子的积等于999999999999000000000001（24位）？
（如无人求解明日公布答案）

太阳 · 发表于 2021-2-26 07:43

数学软件计算:10^60-1=3^3 * 7 * 11 * 13 * 31 * 37 * 41 * 61 * 101 * 211 * 241 * 271 * 2161 * 3541 * 9091 * 9901 * 27961 * 2906161 * 4188901 * 39526741

太阳 · 发表于 2021-2-26 07:54

99999999990000000001=61 * 9901 * 4188901 * 39526741

yangchuanju · 发表于 2021-2-26 16:33

3#（楼）问题答案：
10^18-1=================================
999999999999999999=P1 * P1 * P1 * P1 * P1 * P2 * P2 * P2 * P2 * P5 * P6
P1 = 3
P1 = 3
P1 = 3
P1 = 3
P1 = 7
P2 = 11
P2 = 13
P2 = 19
P2 = 37
P5 = 52579
P6 = 333667
cputime 0:00:00:00
=================================
90909091=P1 * P2 * P2 * P5
P1 = 7
P2 = 13
P2 = 19
P5 = 52579
cputime 0:00:00:00
10^22-1=================================
9999999999999999999999=P1 * P1 * P2 * P2 * P2 * P4 * P4 * P5 * P6
P1 = 3
P1 = 3
P2 = 11
P2 = 11
P2 = 23
P4 = 4093
P4 = 8779
P5 = 21649
P6 = 513239
cputime 0:00:00:00
=================================
9090909091=P2 * P2 * P4 * P4
P2 = 11
P2 = 23
P4 = 4093
P4 = 8779
cputime 0:00:00:00
10^26-1=================================
99999999999999999999999999=P1 * P1 * P2 * P2 * P2 * P3 * P9 * P10
P1 = 3
P1 = 3
P2 = 11
P2 = 53
P2 = 79
P3 = 859
P9 = 265371653
P10 = 1058313049
cputime 0:00:00:04
=================================
909090909091=P3 * P10
P3 = 859
P10 = 1058313049
cputime 0:00:00:00
10^30-1=================================
999999999999999999999999999999=P1 * P1 * P1 * P1 * P2 * P2 * P2 * P2 * P2 * P3 * P3 * P3 * P4 * P4 * P7
P1 = 3
P1 = 3
P1 = 3
P1 = 7
P2 = 11
P2 = 13
P2 = 31
P2 = 37
P2 = 41
P3 = 211
P3 = 241
P3 = 271
P4 = 2161
P4 = 9091
P7 = 2906161
cputime 0:00:00:00
=================================
90909090909091=P1 * P2 * P3 * P3 * P4 * P4
P1 = 7
P2 = 13
P3 = 211
P3 = 241
P4 = 2161
P4 = 9091
cputime 0:00:00:00
10^34-1=================================
9999999999999999999999999999999999=P1 * P1 * P2 * P3 * P4 * P7 * P10 * P11
P1 = 3
P1 = 3
P2 = 11
P3 = 103
P4 = 4013
P7 = 2071723
P10 = 5363222357
P11 = 21993833369
cputime 0:00:00:28
=================================
9090909090909091=P3 * P4 * P11
P3 = 103
P4 = 4013
P11 = 21993833369
cputime 0:00:00:00
=================================

yangchuanju · 发表于 2021-2-26 16:40

4#（楼）问题答案：
10^36-1=================================
999999999999999999999999999999999999=P1 * P1 * P1 * P1 * P1 * P2 * P2 * P2 * P2 * P3 * P4 * P5 * P6 * P12
P1 = 3
P1 = 3
P1 = 3
P1 = 3
P1 = 7
P2 = 11
P2 = 13
P2 = 19
P2 = 37
P3 = 101
P4 = 9901
P5 = 52579
P6 = 333667
P12 = 999999000001
cputime 0:00:00:01
=================================
9900990099009901=P4 * P12
P4 = 9901
P12 = 999999000001
cputime 0:00:00:01
=================================

yangchuanju · 发表于 2021-2-26 16:50

太阳发表于 2021-2-26 07:54
99999999990000000001=61 * 9901 * 4188901 * 39526741

恭喜太阳老师对5楼第一问题解答正确！

5楼第二问题答案：
10^72-1=================================
999999999999999999999999999999999999999999999999999999999999999999999999=P1 * P1 * P1 * P1 * P1 * P2 * P2 * P2 * P2 * P2 * P3 * P3 * P4 * P4 * P5 * P5 * P6 * P8 * P12 * P16
P1 = 3 P3 = 101
P1 = 3 P3 = 137
P1 = 3 P4 = 3169
P1 = 3 P4 = 9901
P1 = 7 P5 = 52579
P2 = 11 P5 = 98641
P2 = 13 P6 = 333667
P2 = 19 P8 = 99990001
P2 = 37 P12 = 999999000001
P2 = 73 P16 = 3199044596370769
cputime 0:00:00:15
=================================
999999999999000000000001=P4 * P5 * P16
P4 = 3169
P5 = 98641
P16 = 3199044596370769
cputime 0:00:00:07
=================================

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