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# How to Explicitly Solve a Thue-Mahler Equation

8. David Hilbert posed a list of mathematical problems in 1900. Hilbert's tenth problem was "Is there an algorithm to determine whether a given Diophantine equation, has a solution with all unknowns taking integer values." Although Matiyasevich Showed that the answer to this problem is negative in the general case; the answer of this problem for a specific diophantine equation $f(x,y)$ that has rational coefficients, is unknown. Thue-Mahler equation is a diophantine equation of the form: $$F(x,y)=c\cdot p_1^{z_1}\cdots p_v^{z_v}$$ where $F$ is homogeneous with integer coefficients, degree $n\geq3$ and $p_1,\ldots p_v$ are distinct rational primes ($v\geq1$). All the unknowns are integer.In 1933, Mahler showed this equation has at most finitely many solutions when $v\in\mathbb{N}$. However, Mahler's proof was non-effective (does not tell anything about an algorithm to find solutions or an upper bound for them). After many tries to solve the Thue-Mahler equation, finally Tzanakis and De Weger explicitly solved it. In this thesis, we will explain this solution. Solving Thue-Mahler equation can help to solve some other kinds of equations and find $S$-points in curves