Numbers of the form 28(10^n – 1)

 
 
Numbers of the form   28 \,(10^{n} - 1)
 
 

28 \,(10^1 - 1) \; = \; 252

252   is a palindrome whose square can be expressed as the product of two reversible numbers in two different ways:

252^2 \; = \; 252\times 252 \; = \; 144\times 441

 

28 \,(10^2 - 1) \; = \; 2772

2772^2 = 7683984 \; = \; 2772\times 2772 \; = \; 1584\times 4851

 

28 \,(10^3 - 1) \; = \; 27972

It is the first palindromic integers containing just the digits 2, 7, and 9, which are divisible by 2, 7, and 9; and, whose square can be expressed as the product of two reversible numbers in two different ways:

27972^2 \; = \; 782432784 \; = \;  27972\times 27972 \; = \; 15984\times 48951

 

28 \,(10^4 - 1) \; = \; 279972

279972^2 \; = \; 78384320784 \; = \; 279972\times 279972 \; = \; 159984\times 489951

 

28 \,(10^5 - 1) \; = \; 2799972

2799972^2 \; = \; 7839843200784 \; = \; 2799972\times 2799972 \; = \; 1599984\times 4899951

 

28 \,(10^6 - 1) \; = \; 27999972

27999972^2 \; = \; 783998432000784
          = \; 27999972\times 27999972 \; = \; 15999984\times 48999951

…………………….

 
 
 
                       ———————————————————-          

2, \; 22, \; 222, \; 2222, \; 22222, \; .....
are integers of the form   2 \, (10^n - 1)/9

9, \; 99, \; 999, \; 9999, \; 99999, \; .....
are integers of the form   (10^n - 1)

2\times 9, \; 22\times 99, \; 222\times 999, \; 2222\times 9999, \; ....
are integers of the form   2 \, (10^n - 1)^2/9
 

2 \, (10^n - 1)^2/9 \; + \; 10^n   are :

(2\times 9) + 10, \; (22\times 99) + 10^2, \; (222\times 999) + 10^3, \; (2222\times 9999) + 10^4, ....

28
2278
222778
22227778
2222277778
222222777778
22222227777778
2222222277777778
222222222777777778
22222222227777777778
…………………….
…………………….

1/9 \; \times \; (2^{n+1} \; 5^n + 1) \,(10^n + 2)
OR    1/9 \; \times \; (2^{2n+1} \; 5^{2n} + 2^n \; 5^{n+1} + 2)

9 \,(2 \, (10^n - 1)^2/9 \; + \; 10^n)
= \; (2^{n+1} \; 5^n \; + \; 1) \,(10^n \; + \; 2)
= \; 2^{2n+1} \; 5^{2n} \; + \; 2^n \; 5^{n+1} \; + \; 2

giving us:

252
20502
2005002
200050002
20000500002
2000005000002
200000050000002
20000000500000002
2000000005000000002
200000000050000000002
…………………….
…………………….

The square of each of these numbers can be expressed as the product of two reversible numbers in two different ways.

                 ——————————————-          

12^2\times 21^2 = 252^2

144\times 441 = 252^2
114444\times 444411 = 225522^2
111444444\times 444444111 = 222555222^2
111144444444\times 444444441111 = 222255552222^2
111114444444444\times 444444444411111 = 222225555522222^2
…………………….
 

144\times 441 = 252^2
144144\times 441441 = 252252^2
144144144\times 441441441 = 252252252^2
144144144144\times 441441441441 = 252252252252^2
144144144144144\times 441441441441441 = 252252252252252^2
…………………….
 

144\times 441 = 252^2
10404\times 40401 = 20502^2
1004004\times 4004001 = 2005002^2
100040004\times 400040001 = 200050002^2

…………………….

                 ——————————————-          

 

                 ——————————————-          

 

                 ——————————————-          
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

4 Responses to Numbers of the form 28(10^n – 1)

  1. pipo says:

    There are more patrons like that:
    9*(22 * 99 + 100) = 20502 and 20502*20502 = 10404 * 40401
    9*(222 * 999 + 1000) = 2005002 and 2005002*2005002 = 1004004 * 4004001
    9*(2222 * 9999 + 10000) = 200050002 and 200050002*200050002 = 100040004 * 400040001

    And some more:
    144 * 441 = 252 ^2
    1584 * 4851 = 2772 ^2
    10404 * 40401 = 20502 ^2
    12544 * 44521 = 23632 ^2
    14544 * 44541 = 25452 ^2
    14884 * 48841 = 26962 ^2
    15984 * 48951 = 27972 ^2
    27648 * 84672 = 48384 ^2
    114444 * 444411 = 225522 ^2
    144144 * 441441 = 252252 ^2
    137984 * 489731 = 259952 ^2
    159984 * 489951 = 279972 ^2
    409739 * 937904 = 619916 ^2
    1022121 * 1212201 = 1113111 ^2
    1042441 * 1442401 = 1226221 ^2
    1256641 * 1466521 = 1357531 ^2
    1004004 * 4004001 = 2005002 ^2
    1050804 * 4080501 = 2070702 ^2
    1024144 * 4414201 = 2126212 ^2
    1141504 * 4051411 = 2150512 ^2
    1214404 * 4044121 = 2216122 ^2
    2044242 * 2424402 = 2226222 ^2
    1044484 * 4844401 = 2249422 ^2
    1154844 * 4484511 = 2275722 ^2
    1323504 * 4053231 = 2316132 ^2
    1236544 * 4456321 = 2347432 ^2
    1266944 * 4496621 = 2386832 ^2
    1444804 * 4084441 = 2429242 ^2
    1354444 * 4444531 = 2453542 ^2
    1440144 * 4410441 = 2520252 ^2
    1454544 * 4454541 = 2545452 ^2
    1585584 * 4855851 = 2774772 ^2
    1599984 * 4899951 = 2799972 ^2
    3066363 * 3636603 = 3339333 ^2
    1376739 * 9376731 = 3592953 ^2
    4048144 * 4418404 = 4229224 ^2
    2632608 * 8062362 = 4607064 ^2
    2763648 * 8463672 = 4836384 ^2

    • benvitalis says:

      Interesting! Note that
      2, 22, 222, 2222, 22222,….. are integers of the form 2*(10^n – 1)/9
      9, 99, 999, 9999, 99999,….. are integers of the form (10^n – 1)

      2*9, 22*99, 222*999, 2222*9999, …. are integers of the form 2*(10^n – 1)^2/9

      2*9 + 10, 22*99 + 10^2, 222*999 + 10^3, 2222*9999 + 10^4, ….

      28,2278,222778,22227778,2222277778

      1/9 (2^(n+1) * 5^n + 1)(10^n + 2)
      OR 1/9 (2^(2n+1) 5^(2n) + 2^n * 5^(n+1) + 2)

  2. pipo says:

    I think there is also a nice patron in:
    27648 * 84672 = 48384 ^2
    2763648 * 8463672 = 4836384 ^2
    276363648 * 846363672 = 483636384 ^2
    Where 27648 = 9 * 3072
    2763648 = 9 * 307072
    276363648 = 9 * 30707072
    84672 = 9 * 9408
    8463672 = 9 * 940408
    846363672 = 9 * 94040408
    Where 48384 = 9 * 5376 (or 84*576)
    4836384 = 9 * 537376 (or 84*57576)
    483636384 = 9 * 53737376 (or 84*5757576)
    etc

  3. pipo says:

    Better:
    27648 = 144*192
    2763648 = 144*19192
    276363648 = 144*1919192
    84672 = 441 * 192
    8463672 = 441 * 19192
    846363672 = 441 * 1919192
    48384 = 252 * 192
    4836384 = 252 * 19192
    483636384 = 252 * 1919192

    pipo

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