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###### Advanced Algorithms Analysis and Design:

They are called Diophantine equations. We want to ask if there is a solution to the positive integer. For example, X2 + y2 = z2 has solutions x = 3, y = 4, and z = 5. So this equation has a solution. On the other hand, x2 + x +2y =39 has no solution. So the question is: Does the given Diophantine equation have a solution? Solving Hilbert’s tenth problem would mean that mathematicians would be able to solve Diophantine equations mechanically. It took almost 70 years to solve this question.

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Hilbert’s tenth problem was the following problem in computer science: We can write a program that takes a Diophantine equation as input and tells us if the equation has a solution in positive integers. He asked about the algorithm.

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Note that understanding what he did is not easy. Matiyašević did not say that he was not smart enough to find a solution. He proved that no such algorithm exists. He showed the impossibility of the existence of an algorithm.

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Finally, Yuri Matiyašević was able to solve Hilbert’s tenth problem. What was Matiyašević’s solution? Did he invent an algorithm? No. Matiyašević proved that Hilbert’s 10th problem is unsolvable. There is no algorithm that can determine whether a Diophantine equation has positive integer solutions.