Using computers to crack open centuries-old mathematical puzzles

In mathematics, no researcher works in true isolation. Even those who work alone use the theorems and methods of their colleagues and predecessors to develop new ideas.

But when a known technique is too difficult to use in practice, mathematicians may neglect important – and otherwise solvable – problems.

Recently, I joined several mathematicians on a project to make one such technique easier to use. We produced a computer package to solve a problem called the “S-unit equation,” with the hope that number theorists of all stripes can more easily attack a wide variety of unsolved problems in mathematics.

Diophantine equations

In his text “Arithmetica,” the mathematician Diophantus looked at algebraic equations whose solutions are required to be whole numbers. As it happens, these problems have a great deal to do with both number theory and geometry, and mathematicians have been studying them ever since.

Why add this restriction of only whole-number solutions? Sometimes, the reasons are practical; it doesn’t make sense to raise 13.7 sheep or buy -1.66 cars. Additionally, mathematicians are drawn to these problems, now called Diophantine equations. The allure comes from their surprising difficulty, and their ability to reveal fundamental truths about the nature of mathematics.

In fact, mathematicians are often uninterested in the specific solutions to any particular Diophantine problem. But when mathematicians develop new techniques, their power can be demonstrated by settling previously unsolved Diophantine equations.

Andrew Wiles, right, receives the Wolflskehl award for his solution of Fermat’s Last Theorem. Peter Mueller/REUTERS
Andrew Wiles’ proof of Fermat’s Last Theorem is a famous example. Pierre de Fermat claimed in 1637 – in the margin of a copy of “Arithmetica,” no less – to have solved the Diophantine equation xⁿ + yⁿ = zⁿ, but offered no justification. When Wiles proved it over 300 years later, mathematicians immediately took notice. If Wiles had developed a new idea that could solve Fermat, then what else could that idea do? Number theorists raced to understand Wiles’ methods, generalizing them and finding new consequences.

No single method exists that can solve all Diophantine equations. Instead, mathematicians cultivate various techniques, each suited for certain types of Diophantine problems but not others. So mathematicians classify these problems by their features or complexity, much like biologists might classify species by taxonomy.

Finer classification

This classification produces specialists, as different number theorists specialize in the techniques related to different families of Diophantine problems, such as elliptic curves, binary forms or Thue-Mahler equations.