 . , , 
  


           .     ,        .    ,   ,    ,   ,      .        -   .     . ,   ,  , .









     .   ,        .   ,   ,    ,             : , ,   .       .           ,     ,    2000   ..  300  ..           .          ,   1700    .               3500   .   ,    ,  ,  ,     ,       .      .             (64 .  ..). ,     ,    .  ,       ,           (  II   ).             .    1000     ,    ,         (, ,  ).

   ,             ,  -.   ,     250   ..,   ,       .  800  ..    .        - -- (  ),   830     -.

 1100                   .        ,     .         .      ,      .      ,  ,        5  !             ,   , ,    .

          .  ,    ,       , ,    . ,    ,      ,        ,  ,  ,      ,   .         ,     ,    ,      , , ,            . ,    ,      ,     .      .              ,     ,     .         ,   .

     . ,      ,          .    .  1866        ,    ,   ,   .     , -   ,     ,   ,     ,      .    ,       ,    ,      .  ,   ,   ,    .       ,      ,      .            ,            .       .          ,    .




  ,  




   ,  - ,   .



     ,   .       ,    ,    .        :   ,        .      ,       .       .        ,      ,         .  ,     ,             .   ,      ,       ,     .   : , , ! ,    , ,  ,         .  ,       ,   ,    ,  .          ,    .

   ,   ,        .       ,            ,        ,    ,   -,        18     .     -  ,  .    ,     .        ,   ,          .   -     ,     , ,  ,           .     :    , ,  ,   !

,   , ,   ,      .

  ( . naturalis  )  ,     : 1, 2, 3, 4, 5, 6, 7, 8,  .    ,    ,    : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,  .

  ,    .  ,     ,   ,        , ,   - .                   .         ,           .      ,     .    ,      ,   ,   ,     ,   ,    .   ,       .     .       ,     ,         .

         : ,      ,  ,    ; ,      ,   ,  .

      ,      .   : 1, 2, 3, 4, 5, 6, 7, 8,                 .           ,        :  ໠ ,  Ỡ    .

     ,   .       ,       ,           ,                      .

             .    ,     ,     .          .         ,      .    ,           .           ,             .     .      ,      .          .       ,     .  ,           .         ,      . 5+3=5+1+1+1,      5  3        .          ,      .         ,        ,      ,    -  .

        ,         .       : 35=3+3+3+3+3.

 ,      ,    ,          : 4


=444.

 ,       ,      ,         ,        .

3


=33=3+3+3=3+1+1+1+1+1+1.

    ,  ,  ,  ,  ,    .        ,         .          ,    .     ,           .      ,    .

  ,    ,       ,    ,        .       ,     a  b      : a<b, a>b, a=b.

,               ,      ,              ,       .     ,  ,    ,         .          .

  .     ,        ,        .        ,      !               .       .             . ,     -      .      .  ,     .    .

 1.    ,     2.   ,    2.

 2.   ,     3   1.      3.

 3.         ,    .

 ,            .    , , 53.      2.  160      1,  80,  40, 20, 10, 5.   2  16.  1: 8, 4, 2, 1.  ,   ,    ,       ,  ,      . ,           ,  ,  .          ,           . ,   ,    :        ,     .     : 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1.  12 ,      .     25,   53,     : 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1.  ,      .    27,    25: 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466,  .  ,   ,   -.         .       ,    ,      25  27?

 -         ,     ,         2       1.     ,   .  2.   ,     3,   1     2.      3.  ,   ,        3,      .             ,    .

            .         (    ,     ),        :  ,   ,  ,   ,   ,     .   ,        ,     :

a+b; a-b; ab; a:b; a


; a<b; a>b; a=b.

  ,     ,      ,     a


.       ,   ,         .      ,      ,        ,   :

2


=8?3


=9; 5


=625?4


=1024.

      : 2


=4


=16.

            ,           :    ,    .       :     .  ,    ,       ,        ,            ,   .

       . ,    ,       ,       ,     ,       .         ,    . ,         ,           .

        ,        ,    .    ,    .    .      ,            .

             , , ,     .

   ,     ,    ,      ,    (    ),   ,      ,     . ,    3  :     3,       3. ,  9   ,      9.         ,          . ,     ,     ,      .     ,   -    .     ,            ,   ,         (     ),            .        ,     .        , ,              ,    . ,        n [25].          .       .           , ,  , .

     (       ),       .      ,  .       ,     ,        ,      .  ,  :








   .             . ,             ,  :




n      , , 


56235=5+6+2+3+5=21.

          :







n       ,







562=5


+6


+2


=25+36+4=65;









n       ,







235=2


+3


+5


=8+27+125=160;









n       ,







12=1+2+3+4+6+12=28;









n      ,







6=1+2+3=6;









n     , 





24=8;









n     , 





30=7;






n       ,




4723=2347;






n       ,




4723=7432;






n     ,




1953=1953=135.

     . -,       ,      ,  ,   .

-,         MicrosoftWord , ,         .

 ,    .

  ,         ,       ,    ,  ,    .      .       ,          ,   ,     ,      . ,    1987652  2,   1+9+8+7+6+5+2=38,  3+8=11 , , 1+1=2.           .     .   ,  ,       : 


n     : 


1987652=2.    : +  ,    ,   ,        .

   : n?9


n=n,         ,    .    :    n        9,    n.

,    9,       ,     :

).            9,       0.

).  ,   0   9,     0,  9.

     ,  ,  -         .      1, 4, 7  9,       2, 3, 7  8.      ,        1, 8  9.

       ,      ,      ,      .         ,      ,     .     .     ,     . ,      ,     ,          .








  .            .   ,        ,   ,         .  ,    .        ,   ,       .    :        .

         ,         ,    ,      ,      .

              ,    ,      ,    ,   .  ,   .            . , 100=(1+2+3+4)


=1


+2


+3


+4


.      ,  .      .

            : , , ,    .  ,        .    88       ,     ,      ,    !   ,     ,  ,   .       .




     





  ,          .



    , ,  ,      ,        .    ,     .  ,       , ,   .           ( ).        : , , , , , , , , , ,   .          ,      . ,     ,     ?    ,    .         ( ),       .

    ,     .       ,              ,      .       .       ,       ,  .




  





     


             :

,    : 1, 2, 3, 4, 5, 6, 8, 9 (  );

,    :  10  99 ( );

,  100  999 ( )   ,     .




   


       ,        ,   ,   , ,   .

     .

    ,     ,    .   ,    2  : 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24,.   ,     2,    1    2: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23,.

       ,  , , .     ,   ,         : 8+18=26, 9+19=28.      ,    .    ,    ,       ,      .             ,    .

-    ,      .  ,        ,    .     ,     2.

    ,   3: 3, 6, 9, 12, 15, 18, ;  4: 4, 8, 12, 16, 20, ;  5: 5, 10, 15, 20,    .   ,   .   12  2, 3, 4, 6  12.   ,       .      24, 48  .         .         . ,    7     7.          .




      


 ,     (   ),  .       . ,     ,      ,       ,   ,        .

  ,    ,    .  ,    ,  .        :     





n,    ,     





n,     . , 





12=1+2+3+4+6+12=28; 





12=1+2+3+4+6=16.

     ,      :

       (





n<n),    ;

       (





n>n),    ;

           (





n=n),    !

 ,   ,      ,      .         ,    .     100   .     ,   ,        .                .




































        6  28.      .

     25.  ,       ,   , ,    74  .             74/25=2,96.

     22,  ,  75.         75/22=3,4(09).   ,   ,  , .           .          1  1000    10 000.       :   168, ,  831    831/168=4,95.

   : 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100  .

    ,     .  ,   ,        !       ,     ?  .     12,      . ,                  ,        945,    386-    .            .        .

1?.             ?

  .  ,           ,        .   .       ,        .   ,         ,   0,2474  0,2480.

      :   ,  28 123,         . ,     ,   .        ,    ,   ,    ,   ,  . -      , ,         .  ,       .

    .   ,  ,      ,  ,   ,   .        : 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50.     ,     .   ,              .   ,    ?  ,         (          ),            .

         ,        .       ,     .        ,       : 6 28.        ,  ,       2





(2


-1),  p      2


-1    .   ,         p=5  p=7.

2


(2


-1)=16(32-1)=1631=496;

2


(2


-1)=64(128-1)=64127=8 128.

         .     (  ).  , ,    ,        .   ,  6,           : 1


+3


+5


+  .     ,       ,  .       ,           .          .     ,       p,   ,    2


-1.     ,     .   XV       33 550 336,  p=13   .     .           : 8589 869 056  137 438 691 328.   p=17  p=19.              .

 ,      6 (           !)      16, 28, 36, 56, 76  96.      6,          1.

     ,   .   2018   50   .  - ,      .        ,     ,    .        ,   .

,    ,   ,  10


;             101.        ,     ,   .

          .          ,        ,     .

     220  284   .            220  284,    .          ?

    220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55  110,    284.

    284: 1, 2, 4, 71  142,    220.

        .          220  284.                 ,      .

 ,    M, N  , : m


+m


++m


=N, n


+n


++n


=M,  m


, m


,,m


    , n


, n


,,n


    N.

     ,          ,      .           850      --   .         : 17 296  18 416; 9363 584  9 437 056.            ,     .             (1601-1665),  ,          - - (1256-1321).           .        ,           .   ,             .

           ,      ,               : 1184  1210,   17 296  18 416      .    1184  1210  ,   ,     1866  16-          .       ,   1184  1210 !  ,   220  284,    ,   .        !

     . ,       .          , -  400, - 1100 ,    ,    2016       .      ,     ,   : 12 285  14 595.    - ,   ,       .     ,      .




    


          .      .     - ,   ,      .         .

   ,      ,         ,  : 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41,  .

   , ,        : 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21,  .

   ,      : 8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42,  .

     .    ,            . , 8=24=222, 12=26=34=223, 18=29=36=233   .




   


   ,    ,    ,        ,    .        ,         ,    ,        ,      .       ,           ,      .








  : 1, 3, 6, 10, 15, 21, 28, 36, 45,  .              : 1, 1+2=3, 1+2+3=6, 1+2+3+4=10, 1+2+3+4+5=15,  .    n-  : P








=(n(n+1))/2.        : P








+P











=(n+1)


.       4: , , , .

,    ,          ,      ,   ,     Worde   .       ,   .

            : 1, 4, 9, 16, 25, 36, 49,  .








          ,             : 1+3=4, 1+3+5=9, 1+3+5+7=16, 1+3+5+7+9=25.    n-   P





=n


.   ,  ,      : 4=1+3, 9=3+6, 16=6+10, P





=P











+P





.        (1808 ):        .  ,    .

      ,      : 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132,  .








      : P











=n(n-1).

   .    : 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210,  .








   n-  : P








=(n(3n-1))/2.

   : 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, .








   n-  : P








=2n


-n.             : P








=P














.

    ,      ,   - .     .

        .           ,    .  .         ,    ,     .        .   ,    3,    6  . .: 1, 1+3=4, 1+3+6=10, 1+3+6+10=20,  .     : 1, 4, 10, 20, 35, 56, 84, 120, 165,  .    : T








=(n(n+1)(n+2))/6.








       ,          .  ,   .

     : 1, 222=8, 333=27, 444=64, 555=125 ,       .








       .                . ,         ,     .             .   :     ,  ,         ,          ,     .        .       ,     .

   ,       , ,           .

  ,      .  ,  ,   .             ,        .  ,            ,    ,  .          ,   ,     .            ,        .      ,         .

 ,   ,      .        (, ),              (, ),      (, , ).  ,         .     ,     : 8P





=P





.      n-   k- .    1637     :      ,       .

     ,   ,     .

     ,        ; ..

        1813 .   ,     : 1637  1813 .




    


        ,      .         .       ,   ( . palindromos   ), ,   ,    (    )      ,   .     , ,  : , , ,   .  ,     ,        ,    , , , , .      (, gig (), eve (), level ()   ),       .              ,     .

       ,   ,  ,    ,  ,    ,    , , 1234567890987654321    .      ,    ?  ,     ,    ,  : ,  ?.    :    ,      ,   .     : 11


=121, 111


=12321, 1111


=1234321.    ,        ,   .    ,    , , 264


=69696, 836


=698896, 2285


=5221225.            .     ,  11


=1331,    ,    ,         .     ,    .  ,        x


  k>4 .   -    .        .         : 11, 101, 131, 151, 181, 191.     !    ,      ,      .  ,           -   ,   .

          .     - - (    ?). -         ,      ( . repdigit  repeated digit   ).   ,   ,      : 11111, 222222, 33333.           ,      ( repeatedunit -  ).      1966   .     Recreations in the theory of numbers: the queen of mathematics entertains.        R


: R


=1, R


=11, R


=111  . .  : 1, 11; 111; 1111; 11111; 1111111  . ,      ,          ,     ,    .       .  ,  -  ,     .      .    ,        .     , ,         .

   ,          .     ,             180 . , 69, 96, 1001.       .       ,       .        0, 1, 8    ,  6  9        180 .        : 1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001, 1111, 1691, 1881,  .

        ,   ,  ,   .       . ,        2  5   .      1961,     1881  1691.

      ,           ,     .  ,    ,   -   ,  0, 1  8,   -  ,   0, 1  8   .    : 1, 8, 11, 88, 101, 111, 181, 808, 818,  .    ,  tetra     .       .                  ,  .

       ,    .       .

  -   ,      .           .           .       JamesJ. Tattersall ElementaryNumberTheoryinNineChapters   , 1999 . , 135   ,   1+3+5=9; 135/9=15.     1  9   .  ,  ,    ,       ,     .    ,      : 11, 12, 15, 24, 36, 111, 112, 115, 128, 132, 135, 144, 175, 212, 216, 224, 312, 315, 384  .

         ,    5        .     5  ,  10,       0, ,     10.   ,       1196342784.          (,    ).

   ,           (   ). , 48=412=86, 672=6112=796=2336.     ,     ()   .       9039  .     ,       .

  : 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 111, 112, 115, 122, 124, 126, 128,  .    ,      . . ,    ,  ,     .




 


         ,   ,   .    ,            .           ,   -    !    .

***

,    ,  ,       ,  1.  , 10    ,  18  ,   18   9=3


.       : 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39,  .

  n       ,                 ,             .

***

    ,     .        ,      ,      10 ( ,   2, 3, 5  7).

   M-,        M.    ,    3-, 5-, 7-  . . . ,  3-  : 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, .            2  3   .  5000     : 2


5


.  5000   5- ,   6-    ,   4-.      ,    2, 3, 5  7, ,    7- .   : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 24, 25, 27, 28, 30, 32, 35, 36, 40, 42,  .  ,      ,      11  .

***

    ,        .  : ,    a


b


,  a  b  .     .           ,         ,    .   : 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200,  .

  ,         ,         .        8  9.   ,      .                  ,            .           .      , , ,      .          : (k+1)


-k


=k


+2k+1-k


=2k+1   .          : (k+2)


-k


=k


+4k+4-k


=4k+4.            ,  ,    . , 2=3


-5


.       6,   ,        .  , 6=25


7


-463


=214 375-214 369.        ,   ,            .

***

   ,                  n.   ,      ,        ,    .      .             ,     ,    ,     .    .    p     p


 .    : 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 26, 28, 29, 31, 33, 34, 35,  .

***

   ( .-.   )  ,       (, , 30=235; ,  30   ).        8. ,  n=pqr,  p, q  r    ,   n : 1, p, q, r, pr, qr, pq, pqr     30  : 1, 2, 3, 5, 6, 10, 15  30.    : 30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195,  .       230 (230=2523)  231 (231=3711).       1309 (1309=71117), 1310 (1310=25131)  1311 (1311=31923).         ,         4.

***

    :    ,            ,       1 (   ),      ,    1.  ,       1,   ,   ,     1,   .     .   ,         ;   ,    . , 19  ,    





19: 1


+9


=82, 8


+2


=68, 6


+8


=100, 1


+0


+0


=1.       143  : 1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100  .                  . ,   19   : 91, 109, 190, 910   .        ,     .

     , ,    ,            ,       1 (   ),      ,    1. ,   1579    : 1







  .


   .

   ,     (https://www.litres.ru/book/vladimir-valentinovich-trosh/naturalnye-chisla-etudy-variacii-uprazhneniya-51357939/chitat-onlayn/)  .

      Visa, MasterCard, Maestro,    ,   ,     ,  PayPal, WebMoney, ., QIWI ,       .


