Sharif Digital Repository

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...