## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

Thursday, October 29, 2015 — 4:00 PM EDT

**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

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.