# Problem 473: Phigital number base
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
representing 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.