Is there a positive integer whose repeat is a square number?

Question

The repeat of a positive integer is obtained by writing it twice in a row. For example, the repeat of 254 is 254254. Is there a positive integer whose repeat is a square number?

Answer 1

This question amounts to 10ⁿa+a=p², where n, a and are positive integers.

So (10ⁿ+1)a=p². This would require 10ⁿ+1 (as well as a) to be a perfect square, and this can never happen, because 11, 101, 1001, 10001 cannot be perfect squares.

If we have a number 10^q+1 then square it we get 10^2q+2×10^q+1, which means there's a 2 between the 1s at each end. For example, 11²=121, 101²=10201, etc. That's missing in 10ⁿ+1, so, no, there are no integer repeats that are perfect squares.