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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 +"

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    "...we shall define a ‘serd’ ..." - who is "we" supposed to be?2011-08-21

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If $p$ is an odd prime and $x^2 = b p^k + y \equiv y \mod p^k$, then $(c p^k + x)^2 \equiv a p^k + y \mod p^{k+1}$ if $2 c x + b \equiv a \mod p$, which can be solved for $c$. So if $y \ne 0$ is a quadratic residue mod $p$, any string of $k$ base-$p$ digits ending in $y$ corresponds to a quadratic residue mod $p^k$.

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    Interesting and amazing solution.2011-08-23