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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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    If you mean what I suspect you mean, the correct spelling is "surd".2011-08-21
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    First of all, thank you for editing.2011-08-21
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    I have given in the last sentance is SERD but not surd.2011-08-21
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    The exact meaning of SERD is: To avoid unnecessary repetition of words we shall define a ‘serd’ as a square ending in repeated, non –zero digits. For example, 144 is an example of a serd.2011-08-21
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    "...we shall define a ‘serd’ ..." - who is "we" supposed to be?2011-08-21

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