Unitary description of the black hole by prime numbers

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.