In my research I often find myself poking around in unfamiliar areas of mathematics. It was a delight and a pleasure to come across one of Hilbert’s famous problems today — the tenth one, concerning Diophantine equations:
Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: to devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.
For those who do not know, a Diophantine equation is one in which only integer solutions are allowed. For example, consider the equation
3 x + 14 y – 8 z = 9.
Normally the set of triples (x, y, z) that satisfy this equation are unlimited; this equation defines a plane in 3-dimensional space and any point on that plane is a valid solution. But the Diophantine restriction makes things trickier, and now we want solutions for which (x, y, z) are integers. If you stare at this example long enough, you can see that x = 1, y = 1, z = 1 is a solution to this system.
I have a slightly more complicated system to solve, so I had to dig a bit deeper than the definitions. Hilbert’s problem asks if there exists an algorithm to solve Diophantine equations — the answer is no, and was proved by Matiyasevich in 1970 in what was apparently an elegant little paper. Today I was reading a 1973 review by Martin Davis, in which section 2 is called “Twenty-four easy lemmas.” Reading these lemmas reminded me why number theory is so confusing; indeed, one needs a 24-step program to get through to the end result.
For what I hear is a nice review of Hilbert’s 10th, see Matiyasevich’s book on the subject.