GNU/Linux factor
Bash tool and python3.
# bash shell
$ factor 101
101: 101
$ factor 1001
1001: 7 11 13
$ factor 10001
10001: 73 137
$ factor 100001
100001: 11 9091
$ factor 1000001
1000001: 101 9901
$ factor 10000001
10000001: 11 909091
$ factor 100000001
100000001: 17 5882353
$ factor 1000000001
1000000001: 7 11 13 19 52579
$ factor 10000000001
10000000001: 101 3541 27961
$ factor 100000000001
100000000001: 11 11 23 4093 8779
$ factor 1000000000001
1000000000001: 73 137 99990001
$ factor 10000000000001
10000000000001: 11 859 1058313049
$ factor 100000000000001
100000000000001: 29 101 281 121499449
$ factor 1000000000000001
1000000000000001: 7 11 13 211 241 2161 9091
$ factor 10000000000000001
10000000000000001: 353 449 641 1409 69857
$ factor 100000000000000001
100000000000000001: 11 103 4013 21993833369
$ factor 1000000000000000001
1000000000000000001: 101 9901 999999000001
$ factor 10000000000000000001
10000000000000000001: 11 909090909090909091
$ factor 100000000000000000001
100000000000000000001: 73 137 1676321 5964848081
$ factor 1000000000000000000001
1000000000000000000001: 7 7 11 13 127 2689 459691 909091
$ factor 10000000000000000000001
10000000000000000000001: 89 101 1052788969 1056689261
$ factor 100000000000000000000001
100000000000000000000001: 11 47 139 2531 549797184491917
$ factor 1000000000000000000000001
1000000000000000000000001: 17 5882353 9999999900000001
$ factor 10000000000000000000000001
10000000000000000000000001: 11 251 5051 9091 78875943472201
$ factor 100000000000000000000000001
100000000000000000000000001: 101 521 1900381976777332243781
$ factor 1000000000000000000000000001
1000000000000000000000000001: 7 11 13 19 52579 70541929 14175966169
$ factor 10000000000000000000000000001
10000000000000000000000000001: 73 137 7841 127522001020150503761
$ factor 100000000000000000000000000001
100000000000000000000000000001: 11 59 154083204930662557781201849
$ factor 1000000000000000000000000000001
1000000000000000000000000000001: 61 101 3541 9901 27961 4188901 39526741
$ factor 10000000000000000000000000000001
10000000000000000000000000000001: 11 909090909090909090909090909091
$ factor 100000000000000000000000000000001
100000000000000000000000000000001: 19841 976193 6187457 834427406578561
$ factor 1000000000000000000000000000000001
1000000000000000000000000000000001: 7 11 11 13 23 4093 8779 599144041 183411838171
$ factor 10000000000000000000000000000000001
10000000000000000000000000000000001: 101 28559389 1491383821 2324557465671829
$ factor 100000000000000000000000000000000001
100000000000000000000000000000000001: 11 9091 909091 4147571 265212793249617641
$ factor 1000000000000000000000000000000000001
1000000000000000000000000000000000001: 73 137 3169 98641 99990001 3199044596370769
$ factor 10000000000000000000000000000000000001
10000000000000000000000000000000000001: 11 7253 422650073734453 296557347313446299
$ factor 100000000000000000000000000000000000001
100000000000000000000000000000000000001: 101 722817036322379041 1369778187490592461
$ factor 1000000000000000000000000000000000000001
factor: ‘1000000000000000000000000000000000000001’ is too large
-
From 101 to 100000000000000000000000000000000000001 there's been only one prime:
101
. -
Most of the
10...01
numbers have more than one divisors. Those that have the least divisors (two) are:- 10001: 73 137
- 100001: 11 9091
- 1000001: 101 9901
- 10000001: 11 909091
- 100000001: 17 5882353
- 10000000000000000001: 11 909090909090909091
- 10000000000000000000000000000001: 11, 909090909090909090909090909091.
-
We notice that the following numbers have divisors with very similar structure:
- 100001: 11 9091
- 10000001: 11 909091
- 10000000000000000001: 11 909090909090909091
- 10000000000000000000000000000001: 11, 909090909090909090909090909091.
-
From the previous observation a questions is raised about the numbers that have divisors
11
and numbers with the structure90..9091
. Is there some underlying pattern? Did an experiment that was illuminating:# python3.x s = "91" t = "90" for i in range(35): r = i*t + s res = 11*int(r) print("11*{}: ".format(r), res, " len(res): {}".format(len(str(res)))) # RESULTS: # -------- 11*91: 1001 len(res): 4 11*9091: 100001 len(res): 6 11*909091: 10000001 len(res): 8 11*90909091: 1000000001 len(res): 10 11*9090909091: 100000000001 len(res): 12 11*909090909091: 10000000000001 len(res): 14 11*90909090909091: 1000000000000001 len(res): 16 11*9090909090909091: 100000000000000001 len(res): 18 11*909090909090909091: 10000000000000000001 len(res): 20 11*90909090909090909091: 1000000000000000000001 len(res): 22 11*9090909090909090909091: 100000000000000000000001 len(res): 24 11*909090909090909090909091: 10000000000000000000000001 len(res): 26 11*90909090909090909090909091: 1000000000000000000000000001 len(res): 28 11*9090909090909090909090909091: 100000000000000000000000000001 len(res): 30 11*909090909090909090909090909091: 10000000000000000000000000000001 len(res): 32 11*90909090909090909090909090909091: 1000000000000000000000000000000001 len(res): 34 11*9090909090909090909090909090909091: 100000000000000000000000000000000001 len(res): 36 11*909090909090909090909090909090909091: 10000000000000000000000000000000000001 len(res): 38 11*90909090909090909090909090909090909091: 1000000000000000000000000000000000000001 len(res): 40 11*9090909090909090909090909090909090909091: 100000000000000000000000000000000000000001 len(res): 42 11*909090909090909090909090909090909090909091: 10000000000000000000000000000000000000000001 len(res): 44 11*90909090909090909090909090909090909090909091: 1000000000000000000000000000000000000000000001 len(res): 46 11*9090909090909090909090909090909090909090909091: 100000000000000000000000000000000000000000000001 len(res): 48 11*909090909090909090909090909090909090909090909091: 10000000000000000000000000000000000000000000000001 len(res): 50 11*90909090909090909090909090909090909090909090909091: 1000000000000000000000000000000000000000000000000001 len(res): 52 11*9090909090909090909090909090909090909090909090909091: 100000000000000000000000000000000000000000000000000001 len(res): 54 11*909090909090909090909090909090909090909090909090909091: 10000000000000000000000000000000000000000000000000000001 len(res): 56 11*90909090909090909090909090909090909090909090909090909091: 1000000000000000000000000000000000000000000000000000000001 len(res): 58 11*9090909090909090909090909090909090909090909090909090909091: 100000000000000000000000000000000000000000000000000000000001 len(res): 60 11*909090909090909090909090909090909090909090909090909090909091: 10000000000000000000000000000000000000000000000000000000000001 len(res): 62 11*90909090909090909090909090909090909090909090909090909090909091: 1000000000000000000000000000000000000000000000000000000000000001 len(res): 64 11*9090909090909090909090909090909090909090909090909090909090909091: 100000000000000000000000000000000000000000000000000000000000000001 len(res): 66 11*909090909090909090909090909090909090909090909090909090909090909091: 10000000000000000000000000000000000000000000000000000000000000000001 len(res): 68 11*90909090909090909090909090909090909090909090909090909090909090909091: 1000000000000000000000000000000000000000000000000000000000000000000001 len(res): 70 11*9090909090909090909090909090909090909090909090909090909090909090909091: 100000000000000000000000000000000000000000000000000000000000000000000001 len(res): 72
From our experiments we can conjecture the following:
- Every number of the type
10...01
with even number of zeroes more than four (6, 8, 10, ...) can be created by multiplying11
with numbers of the type90...9091
. - The number
1001
is being created by multiplying11
with91
. From there on, the rest of such numbers are created by multiplying11
with9091
(100001),909091
(10000001),90909091
(1000000001), etc.
First five examples:
- 1001: 11 * 91 (base case), no of zeroes is 2
- 100001: 11 * 9091 (first case), no of zeroes is 4
- 10000001: 11 * 909091 (second case), no of zeroes is 6
- 1000000001: 11 * 90909091 (third case), no of zeroes is 8
- 100000000001: 11 * 9090909091 (fourth case), no of zeroes is 10
THE PURPOSE OF COMPUTING
IS INSIGHT, NOT NUMBERS