The Red and Black

Issue date: 3/27/08 Section: News

Media Credit: KRISTEN BOYD UC-Berkeley's Bjorn Poonen speaks about number theories at the Cantrell lecture on Wednesday. [Click to enlarge]

In mathematical theories, it often can be difficult to tell whether a solution exists, a University of California Berkeley professor said Wednesday.

Bjorn Poonen, a Berkeley math professor, gave the "Solved and Unsolved Problems in Number Theory" seminar. Poonen is also a four-time winner of the Putnam Undergraduate Mathe-matics Competition, an annual mathematical contest, David Williams, director of the Honors Program, said.

Poonen went over two unsolved examples, the rectangular box problem and the four corners problem, and two solved examples, Fermat's last theorem and Faltings' theorem.

Though technology and heuristics, or methods, help solve theories, Poonen said proofs can be hard for each example.

"Heuristics, used with care, may give qualitatively correct predictions about solutions," he said. "Proving that such predictions are correct can be extremely difficult."

Mathematical research is rewarding to those who solve theories, Poonen said.

Fermat's last theorem

What it is: If the integer n is less than 2, then the equation x^n + y^n = z^n has no solutions in non-zero integers for x, y and z. It was written by mathematician Pierre de Fermat in 1637.
Status: There are at least 10 solutions to this problem, Poonen said. "Solving these may have taken hundreds of years," he said, indicating the long list of people involved in each solution.

Rectangular box problem

What it is: Is there a rectangular box that exists where the lengths of the edges, face diagonals and long diagonals are all rational numbers?

Four corners problem

What it is: Consider a unit square in a plane with corners at (0,0), (1,0), (1,1) and (0,1). Is there a point in the plane where the distance to all four corners is rational?
Status: This problem is still unsolved. "What we want is a simultaneous solution [for the points in this problem]," Poonen said. "I don't expect it to happen ... heuristics say it's unlikely that such points exist."
Status: This problem is still unsolved.

Faltings' theorem

What it is: Once known as the Mordell Conjecture, it is an algebraic curve defined over rational numbers. How many rational points are on the curve?
Status: It was solved in 1983 by German mathematician Gerd Faltings, who stated that the curve has a finite number of points. The theorem was later reproved by Paul Vojta in 1991. But because the theorem is qualitative, Poonen said solving examples of the theorem can be difficult. "Methods exist, but it has not been proven that they always succeed," he said.

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