Gnarly, hundred-year-old mathematical dilemmas get new solutions

A series of unsolved number theory puzzles called Diophantine problems dates back to 3,700 years ago. Over the years, mathematicians have chipped away at them, and recent work has made significant progress on some and shown others to be just as indecipherable as ever.

The researchers used geometry tools to solve the problems, which are named after Diophantus, a 3rd-century Greek mathematician. They consist in determining which solutions exist for polynomial equations such as X^m + Yes^m = z^m. Mathematicians want to know if there are whole or rational solutions to the equations. For example, for X² + Yes² = z², there is an infinity of such solutions.

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

It’s surprising “how little we know about Diophantine geometry compared to other areas of mathematics,” says Bjorn Poonen, mathematician at the Massachusetts Institute of Technology. For example, 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, if the number 114 can be expressed as the sum of three cubes remains an open problem.

The dark side “

For some Diophantine problems, the focus of mathematicians may seem obsessively narrow. Why go to great lengths to determine if 114 can be written as the sum of three cubes? Kiran Kedlaya, a mathematician at the University of California, San Diego, explains that for many simple Diophantine puzzles, “the problem itself is not that central… but the techniques required to solve it are very central.

This property is not uncommon in math. The famous dilemma known as Fermat’s Last Theorem, for example, is also more important because of the techniques developed to solve it than for the problem itself, says Kedlaya, “which does not have many direct consequences for it. number theory. “The tools used to attack it, however, include key advances in algebraic number theory at the end of the 19th century, as well as in modular forms at the start of the 20th century.” Those [developments] are incredibly important in solving many problems in modern number theory, ”he says, including issues related to cryptography.

“The simpler problems tend to be the motivation that gets us to develop techniques that we can then use to solve the problems that really tell us a lot,” he says. For example, Serre’s uniformity problem, which relates to Kedlaya’s research, involves a special type of mathematical curve called a modular curve. However, “the consequences are quite profound and the techniques we use to apply it” to different cases are rooted in previous work on Fermat’s problem, notes Kedlaya.

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

New tools for old problems

Rather than using tools from geometry and other fields to solve specific Diophantine problems, it would have been possible to develop computer programs to solve the general case of such problems. But mathematicians Martin Davis, Yuri Matiyasevich, Hilary Putnam, and Julia Robinson have shown that finding complete solutions to these problems isn’t as easy as tasking a computer to research them. Their work resulted in a 1970 theorem which answered the German mathematician David Hilbert‘s famous 10th problem. This problem focused on finding an algorithm to determine whether, for a system of polynomial equations with integer coefficients, there is a solution in the integers, notes Kedlaya. Thinking that such an algorithm could be found, “Hilbert was an optimist,” Kedlaya says. “Hilbert was determined to try to deal with general categories of problems.”

But Matiyasevich’s theorem, also called DPRM theorem or MRDP theorem, showed that such an algorithm does not exist. The finding means that “the general problem of this type is, of course, unsolvable” and that individual instances of these problems can be “very difficult to solve,” Kedlaya explains.

Oddly, Corwin notes that for polynomial equations (or systems of such equations) with many variables, no one knows if an algorithm can be found to determine whether rational solutions exist. “It’s up to everyone to guess,” he said. Poonen worked to show that such a general method of finding solutions in rational numbers is impossible.

For some of these old questions, including those posed by Diophantus himself, “we are only now developing methods that can help answer them,” says Jennifer Balakrishnan, a mathematician at Boston University. For example, a problem from the book of Diophantus Arithmetic concerns the question of whether positive and rational solutions X and Yes exists such that the equation Yes² = X⁸ + X⁴ + X² is satisfied. Although Diophantus provided a solution, which is X = ½ and Yes = ⁹ ?? ₁₆, Balakrishnan says that until 1998, it was not known how many other solutions existed. In a doctoral thesis at the University of California at Berkeley, Joseph Wetherell presented techniques to answer this question.

More recently, Balakrishnan and his collaborators have developed new techniques to find similar solutions. A recent impactful result, she says, has been that of Brian Lawrence and Akshay Venkatesh new evidence of what is called the Mordell conjecture. Although Gerd Faltings first proved the Mordell conjecture in 1983, the work of Lawrence and Venkatesh “gives another perspective on a problem that is almost 100 years old,” Balakrishnan says. These and other advances show that interest in Diophantine geometry has grown in recent years, says Corwin, “particularly with the rise of new methods.”