Suppose there is a repeating string in the sequence of digits in a and k0 is the first number that k0^2 if fully inside the repeating region. Then:

 

1. It is easy to show that (k+1)^2 can be only 1 digit longer than k^2.

2. Therefore, there is always k>k0, that the length of k^2 is a multiple of the length of the repeating string

3.  If (k+1)^2 has the same length as k^2, then a repeating pattern is not possible. If (k+1)^2 is one digit longer than k^2, then (k+1)^2=10*k^2+m, where m<10, which again is not possible.

 

A similar proof can be offered for c.