I am working on square ending in repeated digits in different bases. I have encountered the following problems during my work. can you generalize the following??? If the digit $a < p$ is a quadratic residue $\pmod p$, then the base $p$ number $A_n$ consisting of $n$ $a$’s is a quadratic residue $\pmod {p^n}$ and as a corollary, squares exist in base $p$ ending in $n$ $a$’s.
Also, generalize "A serd ending 4444 is not possible in any base of the form +"