  In this paper, we study the thermofield double states of doubly-holographic
gravity in two copies of the horizons. We show that the asymptotically AdS
spacetimes describe an entangled states of a pair of CFTs based on the Farey
sequence. We propose a new technique to geometrize the black hole horizon. Our
protocol is based on the so-called Farey diagram. We construct states and
entropies to describe the unit cells on the horizon. As a result, we have
proved that the quantum states on the horizon are encoded by prime numbers.
Therefore, we found that the entropy of the code space and area law are
writtens in logarithmic form of the prime numbers. We show that the number of
connected components of the Farey sequence can build the Fermi--Dirac
distribution. To solve the information paradox problem, we find that the
Hawking radiation follows geodesic of the Farey diagram, then he turns around
and falls on the horizon. Our aim is to show that there is appearance of
several Page times, because of discontinuous emission of these radiations.
Finally, we mention the possibility to describe the quantum Hall effect by
using the Farey diagram. In this paper, we find a link between quantum
information and the theory of numbers passing through geometry.
