Prefix of Fibonacci number Given some prefix how can we check if this prefix belongs to a Fibonacci number? If yes then to which one?
By the prefix of number I define first $n$ digits.
For example
10 is prefix of 10231
1234 is prefix of 1234592

and so on.
I was trying to get anything from Binet's formula, but couldn't... Thanks for any tip.
 A: Your idea of using the Binet Formula can be made to work. 
If for arbitrarily large $n$,  $\frac{\varphi^n}{\sqrt{5}}$ has any specified prefix $P$, then for arbitrarily large $n$, $F_n$ has  any specified prefix $Q$.  (If we are worried about a prefix like $2000$, change it to $20001$. If $\frac{\varphi^n}{\sqrt{5}}$ has $20001$ as a prefix, and $n$ is large, then $F_n$ has $2000$ as a prefix.) 
Now we show that for any $P$, there are infinitely many $n$ such that $\frac{\varphi^n}{\sqrt{5}}$ has prefix $P$. Consider $\log_{10}\frac{\varphi^n}{\sqrt{5}}=n\,\log_{10}\varphi -\log_{10}\sqrt{5}$.
It is easy to show that $\log_{10}\, \varphi$ is irrational. Thus the fractional parts of $n\,\log_{10}\varphi$, as $n$ ranges over the natural numbers, are dense in the unit interval, and therefore so are the fractional parts of $n\,\log_{10}\varphi -\log_{10}\sqrt{5}$. The result follows.
Comment: The above solution gives no real information about the "if so which one" part of the question. There are infinitely many. One would like to produce a decent upper bound for the smallest one.  Calculations I have made with continued fractions for very similar problems, such as producing a specified prefix for $2^n$, have not given good bounds.
