A series of unsolved puzzles in number theory called Diophantine problems date back to 3,700 years ago. Over the years mathematicians have whittled away at them, and recent work has made significant progress on some—and showed others to be just as uncrackable as ever.

Researchers have been using tools from geometry to tackle the problems, which are named after Diophantus, a third-century Greek mathematician. They involve determining which solutions exist for polynomial equations such as xn + yn = zn. Mathematicians aim to find out if there are any integer or rational solutions to the equations. For instance, for x2 + y2 = z2, infinitely many such solutions exist.  

Diophantine geometry is the field of math focused on the relationship between the number theory properties of an equation, such as its rational or integer solutions, and “the geometric properties, like the topology of the set of complex solutions to the equation,” says David Corwin, a mathematician at Ben Gurion University of the Negev in Israel.

It is surprising “how little we know about Diophantine geometry compared to other fields in mathematics,” says Bjorn Poonen, a mathematician at the Massachusetts Institute of Technology. For instance, he notes that although mathematicians know that the number 20 can be written as the sum of three cubes, as in 33 + 13 + (–2)3 = 20, whether the number 114 can be expressed as the sum of three cubes remains an open problem.

The “Dark Side”

For some Diophantine problems, mathematicians’ focus might seem obsessively narrow. Why expend substantial effort to determine whether 114 can be written as the sum of three cubes? Kiran Kedlaya, a mathematician at the University of California, San Diego, says that for many Diophantine puzzles that are simple to state, “the problem itself is not so central … but the techniques that are needed to solve it are very central.”

This property is not uncommon in math. The famous quandary known as Fermat’s Last Theorem, for instance, is also more important because of the techniques developed to solve it than for the problem itself, Kedlaya says, “which doesn’t have much in the way of direct consequences for number theory.” The tools used to attack it, however, include key advances in algebraic number theory in the late 19th century, as well as in modular forms in the early 20th century. “Those [developments] are incredibly important for solving lots of problems in modern number theory,” he says, including questions connected to cryptography.

“The very simplest problems tend to be the motivation that lead us to develop techniques that we can then use to solve the problems that really tell us a lot,” he says. For example, the Serre Uniformity Problem, which relates to Kedlaya’s research, concerns a special type of mathematical curve called a modular curve. However, “the consequences of it are quite deep and the techniques that we’re using to apply it” to different cases are rooted in earlier work on the Fermat problem, Kedlaya notes.

Still, some of the Diophantine problems are more intractable than others. “Many researchers in the field try to develop new methods for solving Diophantine equations,” Poonen says, “but I work also on the ‘dark side’ by trying to prove that some classes of problems are unsolvable.”

New Tools for Old Problems

Rather than employ tools from geometry and other fields to solve specific Diophantine problems, it might have been possible to develop computer programs to solve the general case of such problems instead. But mathematicians Martin Davis, Yuri Matiyasevich, Hilary Putnam and Julia Robinson, showed that finding full solutions to these problems isn’t as simple as tasking a computer with searching for them. Their work culminated in a 1970 theorem that answered German mathematician David Hilbert’s famous 10th problem. That problem focused on finding an algorithm for determining whether, for some system of polynomial equations with integer coefficients, there exists a solution in the integers, Kedlaya notes. In thinking that such an algorithm could be found, “Hilbert was an optimist,” Kedlaya says. “Hilbert was big on trying to take care of general classes of problems.”

But the Matiyasevich theorem, which is also called the DPRM theorem or the MRDP theorem, showed that such an algorithm doesn’t exist. The discovery means that the “general problem of that type is, of course, intractable,” and individual instances of these problems can be “very hard to solve,” Kedlaya says.

Curiously, Corwin notes that for polynomial equations (or systems of such equations) in multiple variables, no one knows whether an algorithm can be found for determining whether rational solutions exist. “It’s anyone’s guess,” he says. Poonen has worked to show that such a general method for finding solutions in rational numbers is impossible.

For some of these ancient questions, including ones posed by Diophantus himself, “we’re only just now developing methods that can help answer them,” says Jennifer Balakrishnan, a mathematician at Boston University. For example, a problem from Diophantus’s book Arithmetica concerns whether positive, rational solutions for x and y exist such that the equation y2 = x8 + x4 + x2 is satisfied. Though Diophantus provided a solution, which is x = ½  and y = 916, Balakrishnan says that until 1998, it was unknown how many other solutions existed. In a doctoral thesis at the University of California, Berkeley, Joseph Wetherell presented techniques for answering this question.

More recently, Balakrishnan and her collaborators have been developing new techniques to find similar solutions. A recent impactful result, she says, was Brian Lawrence and Akshay Venkatesh’s new proof of something called Mordell’s conjecture. Though Gerd Faltings first proved Mordell’s conjecture in 1983, the work by Lawrence and Venkatesh “gives another perspective on a problem that's nearly 100 years old,” Balakrishnan says. These and other advances show that interest in Diophantine geometry has been growing in recent years, Corwin says, “especially with the rise of new methods.”