Question 21
We call a number doubly square if (1) it is a square with an even number $2n$ of (decimal) digits, (2) its first $n$ digits form a square, (3) its last $n$ digits form a non-zero square. For example, $1681$ is doubly square, but $2500$ is not.
(1) Find all $2$-digit and $4$-digit doubly square numbers.
(2) Is there a $6$-digit doubly square number?
(3) Show that there is a $20$-digit doubly square number.
(4) Show that there are at least ten $100$-digit doubly square numbers.
(5) Show that there is a $30$-digit doubly square number.
(1) Find all $2$-digit and $4$-digit doubly square numbers.
(2) Is there a $6$-digit doubly square number?
(3) Show that there is a $20$-digit doubly square number.
(4) Show that there are at least ten $100$-digit doubly square numbers.
(5) Show that there is a $30$-digit doubly square number.