Numbers of the form

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

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:

…………………….

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are integers of the form

are integers of the form

are integers of the form

are :

…………………….

…………………….

OR

giving us:

…………………….

…………………….

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

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…………………….

…………………….

…………………….

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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

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)

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

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