How to see the list


When you open the list , you can see as below on the top of it ;
    1    no prime factors
    2    11
    3    3  37
    4    (2)  101
    5    41  271
    6    (2,3)  7  13
    7    239  4649
    8    (2,4)  73  137
    9    (3)  3  333667
   10    (2,5)  9091
   11    21649  513239
   12    (2,3,4,6)  9901
   13    53  79  265371653
   14    (2,7)  909091
   15    (3,5)  31  2906161
   16    (2,4,8)  17  5882353
   17    2071723  5363222357
   18    (2,3,6,9)  19  52579
   19    1111111111111111111
   20    (2,4,5,10)  L  M
    L    3541
    M    27961
   21    (3,7)  43  1933  10838689
   22    (2,11)  11  23  4093  8779
   23    11111111111111111111111
   24    (2,3,4,6,8,12)  99990001
   25    (5)  21401  25601  182521213001


Here are 6 examples of the different cases .
" Rn " denotes the repunit number of n digits ; Rn = 111...11 (n digits)

Example 1 : R5 = (105-1) / 9

Please search ;
    5    41  271

"41   271" means 41 and 271 are R5 's original non trivial divisors . As 5 is prime number , R5 has no trivial factors . So the list indicates only its nontrivial factors , 41 and 271 .
Thus the complete factorization of R5 is ;

R5 = 41 * 271

Example 2 : R12

Please search ;
   12    (2,3,4,6)  9901
This time "(2,3,4,6)" is added before the nontrivial factor of R12 , 9901 ."(2,3,4,6)" means R12 has original non trivial divisors of R2 , R3 , R4 and R6 .
To obtain the complete factorization of R12 , you need to see ;
R2 :     2    11
R3 :     3    3  37
R4 :     4    (2)  101
R6 :     6    (2,3)  7  13
Thus the complete factorization of R12 is ;
R12 = 11 * 3 * 37 * 101 * 7 * 13 * 9901
              = 3 * 7 * 11 * 13 * 37 * 101 * 9901

Example 3 : R20

Please search ;

   20    (2,4,5,10)  L  M
    L    3541
    M    27961
This means R20 is the product of nontrivial divisors of R2 , R4 , R5, R10 , L and M .
 R2 :     2    11
 R4 :     4    (2)  101
 R5 :     5    41  271
R10 :    10    (2,5)  9091
Thus the complete factorization of R20 is ;
 R20= 11 * 101 * 41 * 271 * 9091 * 3541 * 27961
          = 11 * 41 * 101 * 271 * 3541 * 9091 * 27961

Example 4 : R191

Please search ;

  191    4473297929  112103021940812743353521
         2215707086722579688559276358532826248956997296824658187129019075996613880040041515650182700179703274  $
         5294740404832373351030284459396524271152620265237784647679
R191 has 3 non-trivial factors ;
1 : 4473297929
2 : 112103021940812743353521
3 : 22157070867225796885592763585328262489569972968246581871290190759966138800400415156501827001797032745294740404832373351030284459396524271152620265237784647679
But the 3rd factor is too long to fit in 1 line , it use 2 lines." $ " shows that the factor continues to the next line.

Column length



  191    4473297929  112103021940812743353521
         2215707086722579688559276358532826248956997296824658187129019075996613880040041515650182700179703274  $
         5294740404832373351030284459396524271152620265237784647679
         <-----------------------------             column  length             ----------------------------->
There are 3 types of column lengths ; 60 , 80 , and 100 in the compressed forms .
For example , you can use them ,
column length = 60 : for printing the list on B5 sized paper
column length = 80 : for printing the list on A4 sized paper
column length = 100 : for browsing the list on PC

Example 5 : R323

Please search ;

  323    (17,19)  851908127328669427                                                                           c 271
"c 271" means R323 has a 271 digit composite cofactor whose factorization is unknown now .
Thus , the current factorization of non-trivial part of R323 is ;
851908127328669427  * (composite number of 271 digits) .

Example 6 : R2503

When you search the factorization of this in the list , you will see as below ;

 2502    (2,3,6,9,18,139,278,417,834,1251)  1203463  516041703361                                              c 811
 2504    (2,4,8,313,626,1252)  34810609  31894683299354153                                                     c1224

The list does not contain the result of the factorization of R2503 .
In fact , R2503 has the non-trivial part of 2503 digits .
So R2503 is not in the list .