Let A1 = 14, and let An be the number of the form 1444... with n fours following the 1. Find all n such that An is a perfect square.   It is obvious that A2 = 144 is a perfect square and possibly the only one, yet I haven't found a way to prove that.  Possibly the biggest hint is that if $$ An = (13*10^n - 4) / 9\\$$ and if $$ An = k^2 $$  then $$ k = 13i + 1 \\ or \\ k = 13i + 12 $$ for some value of i.
We know that $12^2 = 144$ and that $38^2 = 1444$. Are there any other perfect squares in the form of $\frac{13}{9} (10^n - 1) + 1$ (i.e. $1$ followed by $n$ $4$'s), and how would we prove it?
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