David Hilbert published twenty-three problems that contributed to the development of mathematics in the 20th century. Hilbert’s tenth problem asked to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers. In 1970, Yuri Matijasevich finally proved the missing link in decades-long proof research to show that such a process, today regarded as an algorithm, does not exist. This work contains proof of unsolvability to the Tenth problem. In the beginning, we indicate and clarify the basic notions in the theory of computability. We introduce the diophantine set for which we want to demonstrate the equivalence to the semidecidable set established in the theory of computability. Diophantine equations, i.e., equations composed with integer coefficients and solutions, define diophantine sets. Proving that the diophantine sets are also semi-decidable implies the unsolvability of Hilbert’s tenth problem. Furthermore, we also show that the exponential functions are diophantine using Fibonacci’s sequences, which was the last obstacle of the mentioned equivalence. Introducing the universal diophantine equation, as well as the Turing machines makes it possible to conclude the equivalence. For the proof, we also introduce different types of coding, e.g., Cantor’s and Gödel’s coding, and necessary diophantine relations.