**
Mohammad
Mahmoud,
Department
of
Pure
Mathematics,
University
of
Waterloo**

“Hilbert’s 10th Problem”

In 1900, the German mathematician David Hilbert outlined 23 major mathematical problems to be studied in the coming century. His ”questions” ranged greatly in topic and precision. They were designed to serve as examples for the kinds of problems whose solutions would lead to the furthering of disciplines in mathematics.

We are going to talk about one of his interesting problems which is no. 10 : Given a Diophantine equation with any number of unknowns and with integer coefficients, devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in integers.

(A Diophantine equation is an equation of the form P (x1, . . . , xm) = 0 where P is a polynomial.)

The words ”process” and ”algorithm” had no rigorous mathematical meaning until the work of G ̈odel, Turing, Post, Church and other logicians in the 1930s. It took mathematicians 70 years to show that Hilbert was too optimistic and that such a process is impossible; there is no such algorithm. The impossibility of obtaining a general solution was proven by the 23 year old Yuri Matiyasevich in 1970. He contributed the last young smart lines to earlier awesome work by Martin Davis, Hilary Putnam, and Julia Robinson.

Let’s see what these people have done, and maybe we can say something about the yet unsolved version of the problem over the rationals!

MC 5417