def problem473():
    """


    Let $\\varphi$ be the golden ratio: $\\varphi=\\frac{1+\\sqrt{5}}{2}.$
       Remarkably it is possible to write every positive integer as a sum of
       powers of $\\varphi$ even if we require that every power of $\\varphi$ is
       used at most once in this sum.
       Even then this representation is not unique.
       We can make it unique by requiring that no powers with consecutive
       exponents are used and that the representation is finite.
       E.g: $2=\\varphi+\\varphi^{-2}$ and $3=\\varphi^{2}+\\varphi^{-2}$

       To represent this sum of powers of $\\varphi$ we use a string of 0's and
       1's with a point to indicate where the negative exponents start.
       We call this the representation in the phigital numberbase.
       So $1=1_{\\varphi}$, $2=10.01_{\\varphi}$, $3=100.01_{\\varphi}$ and
       $14=100100.001001_{\\varphi}$.
       The strings representing 1, 2 and 14 in the phigital number base are
       palindromic, while the string representating 3 is not.
       (the phigital point is not the middle character).

       The sum of the positive integers not exceeding 1000 whose phigital
       representation is palindromic is 4345.

       Find the sum of the positive integers not exceeding $10^{10}$ whose
       phigital representation is palindromic.
    """
    pass
