Halting Problem for One-State Turing Machines

In [1], Turing has established the well-known result of the indecidability of the general halting problem of Turing machines. It is also well known that there exists entire classes of Turing machines whose halting is decidable (machines operating on an alphabet of one symbol, machines with finite tapes etc ...). In this article we are paying attention on Turing machines with a single state for which we prove that the halting is decidable. The interest of this problème is that it settles exactly with other works the limit between decidable and undecidable in regard of the number of internal states of the machine