In 1859, Riemann, who had been a student of Gauss, took up the question of the distribution of primes in his only paper on number theory. With that paper he revolutionized the field, as he had the fields of geometry (his math became the basis for Einstein's theory of gravitation) and several other branches of mathematics. What Riemann discovered was a way of using the properties of a relatively simple function to count the primes.

What was so remarkable about Riemann's zeta function was that it somehow took a question about prime numbers -- those discrete atoms of simple arithmetic, things easy to imagine -- and put it in terms of a far larger and more esoteric class of numbers known as complex numbers. Complex numbers are a generalization of the familiar decimal numbers that mathematicians call the real numbers.

While the real numbers can be thought of as points on an infinite line, the complex numbers are points on a plane. One axis of this complex plane corresponds to the real numbers, and the other corresponds to the ''imaginary'' numbers -- which were introduced so that negative numbers could have square roots, and are no more imaginary than real numbers. A function like Riemann's zeta function is simply a rule that takes a point on this plane and sends it to some other point.

By moving the problem to the complex plane Riemann had access to a whole new set of powerful mathematical tools, many of which he had developed himself. What was going on with the primes turned out to be a shadow of what was going on in this more general world.

Riemann showed that if he knew where the value of his zeta function went to zero he would be able to predict the distribution of the primes. He was able to prove that aside from some ''trivial'' zeros -- located at -2, -4, -6, and so on and thus easily included in his equations -- the zeros of the zeta function all lay within a strip one unit wide running along the imaginary axis.

Somehow the distribution of these zeros mirrored or encoded the distribution of the prime numbers. Riemann guessed that all of the zeros ran along the middle of the critical strip like the dotted line on a highway. Nobody is sure why he made this guess, but it has proven to be inspired. Over the past few decades billions of zeros of the zeta function have been calculated by computer, and every one of them obeys Riemann's hypothesis.

Most of the conference attendees would be shocked if a stray zero were found and Riemann was proven wrong. They would agree with John Frye, the chief executive of Frye's Electronics and a math major who used his fortune to found the mathematics institute. ''I think we would have a better chance of finding life on Mars than finding a counter-example,'' he said.