## 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.

——————————————-

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

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

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

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

…………………….

——————————————-

——————————————-

——————————————-

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