Mathematicians find new solutions to an ancient puzzle

Inspired by the accomplishments of the 22-year-old graduate student, Jacobi began working on mathematics as a hobby after he retired from the defense industry in 1989.

Fortunately, this was not the first time he had dealt with Diophantine equations. He was familiar with them because they are commonly used in physics for calculations relating to string theory.

Jacobi started searching for new solutions to the puzzle using methods he found in some number theory texts and academic papers.

He used those resources and Mathematica, a computer program used for mathematical manipulations.

Jacobi initially found a solution for which each of the variables was 200 digits long. This solution was different from the other 88 previously known solutions to this puzzle, so he knew he had found something important.

Jacobi then showed the results to Madden. But Jacobi initially miscopied a variable from his Mathematica computer program, and so the results he showed Madden were incorrect.

“The solution was wrong, but in an interesting way. It was close enough to make me want to see where the error occurred,” Madden said.

When they discovered that the solution was invalid only because of Jacobi’s transcription error, they began collaborating to find more solutions.

Madden and Jacobi used elliptic curves to generate new solutions. Each solution contains a seed for creating more solutions, which is much more efficient than previous methods used.

In the past, people found new solutions by using computers to analyze huge amounts of data. That required a lot of computing time and power as the magnitude of the numbers soared.

Now people can generate as many solutions as they wish. There are an infinite number of solutions to this problem, and Madden and Jacobi have found a way to find them all.

The title of their paper is, “On a⁴ + b⁴ +c⁴ +d⁴ = (a + b + c + d)⁴."

“Modern number theory allowed me to see with more clarity the implications of his (Jacobi’s) calculations,” Madden said.

“It was a nice collaboration,” Jacobi said. “I have learned a certain amount of new things about number theory; how to think in terms of number theory, although sometimes I can be stubbornly algebraic.”

Contact information:

Daniel Madden, 520-621-4665

Related Web sites:
UA Mathematics Department
http://math.arizona.edu/