9TH DIGIT PALINDROMIC

[MATHPRITOM]

Find the smallest palindromic number of 9 digits whose root is also a palindromic.

[kfoozminus]

$10001^2=100020001$

[nafistiham]

$10001^2=100020001$

Probably, that is correct in this way,

the smallest $3$ $9$ digit palindrome are $100000001,100010001,100020001$
first $2$ are not squares.So....

[Actually, I wondered, how it is the smallest. So, shared my observation.]

[Fahim Shahriar]

Though it's already done with nafis tiham's observation, I'd like to give a solution.

Suppose the root of the palindrome is $10000a+1000b+100c+10b+a = 10001a+101b+100c$

Square it, and take (mod 4). We will get
$a^2+b^2+ab=x^2 (mod 4)$
[We know $x^2=0 or 1 (mod 4)$]

$a$'s least value is 1(it's leftmost digit) and $b$ is either 0 or an even number and its least value is 0. For this $c \leq 2$ and least value 0.

So the root is 10001. And the number is 100020001.

[kfoozminus]

the smallest $3$ $9$ digit palindrome are $100000001,100010001,100020001$
first $2$ are not squares.So....

[Actually, I wondered, how it is the smallest. So, shared my observation.]

$9999^2$ is a $8$ digit number, $10000^2$ isn't palindromic, so...