Palindromic squares and their roots

What if a number multiplied with itself (squared) became a palindromic number? This is the case for some special numbers, and they are called palindromic squares.

The first twenty such numbers are as follows:

1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944, 69696, 94249, 698896, 1002001, 1234321, 4008004, 5221225, 6948496, 100020001

Do you see anyone you like? Are they not pretty and pleasent to look at? This is not something I have made up, they do have an entry in the OEIS (The On-Line Encyclopedia of Integer Sequences), filed under A002779.

Each of them is the product of a number squared and are palindrome:

121 is palindromic square, since 11^2 = 121
484 is palindromic square, since 22^2 = 484
676 is palindromic square, since 26^2 = 676
10201 is palindromic square, since 101^2 = 10201

The list of the first twenty palindromic squares with their square roots:

Palindromic squareSquare root11429312111484226762610201101123211111464112140804202449442126969626494249307698896836100200110011234321111140080042002522122522856948496263610002000110001

Now, if you are thinking what I’m thinking … can we only include the numbers where the factor also is palindromic? Yes we can:

Palindromic squarePalindromic square root1142931211148422102011011232111114641121408042024494421210020011001123432111114008004200210002000110001