Four researchers have recently come out with a model that upends the conventional wisdom in their field. They have used intensive computational data to suggest that for decades, if not longer, prevailing opinion about a fundamental concept has been wrong.

These are not biologists, climatologists or physicists. They don’t come from a field in which empirical models get a say in determining what counts as true. Instead they are mathematicians, representatives of a discipline whose standard currency — indisputable logical proof — normally spares them the kinds of debates that consume other fields. Yet here they are, model in hand, suggesting that it might be time to re-evaluate some long-held beliefs.

The model, which was posted online in 2016 and is forthcoming in the Journal of the European Mathematical Society, concerns a venerable mathematical concept known as the “rank” of an algebraic equation. The rank is a measurement that tells you something about how many of the solutions to that equation are rational numbers as opposed to irrational numbers. Equations with higher ranks have larger and more complicated sets of rational solutions.

Since the early 20th century mathematicians have wondered whether there is a limit to how high the rank can be. At first almost everyone thought there had to be a limit. But by the 1970s the prevailing view had shifted — most mathematicians had come to believe that rank was unbounded, meaning it should be possible to find curves with infinitely high ranks. And that’s where opinion stuck even though, in the eyes of some mathematicians, there weren’t any strong arguments in support of it.

“It was very authoritarian the way people said it was unbounded. But when you looked into it, the evidence seemed very slim,” said Andrew Granville, a mathematician at the University of Montreal and University College London.

Now evidence points in the opposite direction. In the two years since the model was released, it has convinced many mathematicians that the rank of a specific type of algebraic equation really is bounded. But not everyone finds the model persuasive. The lack of resolution raises the kinds of questions that don’t often attend mathematical results — what weight should you give to empirical evidence in a field where all that really counts is proof?

“There is really no mathematical justification for why this model is exactly what we want,” said Jennifer Park, a mathematician at Ohio State University and a co-author of the work. “Except that experimentally, a lot of things seem to be working out.”

Point to Point

If you’re handed an equation, you can graph its solutions and produce a curve. Mathematicians want to know how many of these solutions are rational numbers — values that can be expressed as a ratio of two integers (such as 1/2, −3, or 4483/929).

Rational solutions are hard to find systematically, but mathematicians have techniques that work under some circumstances. Say you have the equation x2 + y2 = 1. The graph of the solutions to this equation form a circle. To find all the rational points on that circle, start with one particular rational solution — say, the point on the circle where x is 1 and y is 0. Then draw a line through that point that intersects the circle at one other point. So long as the slope of your line is rational, the second point of intersection will also be a rational solution. Through the line-drawing procedure, you’ve parlayed one rational solution into two.

And there’s no reason to stop there. Repeat the procedure, drawing a line with a different rational slope through your second rational point — that line will intersect the circle at a third rational point. If you keep doing this forever, you’ll eventually find all of the infinitely many rational points on the circle.

In the case of the circle, you only need to start with one rational point to find them all. The number of rational solutions you need to know at the outset in order to uncover the rest is known as the “rank” of a curve. The rank is a tidy way of characterizing an infinite set of rational solutions with just a single number. “It’s sort of the best possible way of describing rational solutions for these curves,” said Bjorn Poonen, a mathematician at the Massachusetts Institute of Technology and a co-author of the model along with Park, John Voight of Dartmouth College, and Melanie Matchett Wood of the University of Wisconsin, Madison.

The circle is a quadratic equation, also known as a degree-two equation. (“Degree” refers to the value of the equation’s highest exponent.) Mathematicians have had a complete understanding of how to find rational solutions to degree-two equations for more than a century.

The next type of equation is the elliptic curve, which features a variable raised to the third power. Elliptic curves exist in a sweet spot of mathematical inquiry. They’re more complicated than degree-two equations, which makes them interesting to study, but they’re not too complicated. A modified form of the line-drawing procedure still applies to elliptic curves, but it ceases to work with equations of degree four and higher.

Elliptic curves come in a variety of ranks. With some elliptic curves, for instance, you could start with one rational point, apply the line-drawing procedure, and fail to uncover all rational points. You might need to be given a second rational point, wholly unrelated to the first. Then you’d start a fresh line-drawing procedure from that second point to uncover the balance of the rational points. A curve for which you need to know two rational points at the outset in order to find all the rational points has rank two.

There’s no proven limit to how high the rank of an elliptic curve can be. The higher the rank of an equation, the vaster and more intricate the curve’s set of rational solutions. “Rank is somehow measuring how complicated the set of solutions is,” Poonen said.

Yet rank has eluded mathematicians’ efforts to encapsulate it in a theory. If you’re handed an elliptic curve, there’s no obvious relationship between what the curve looks like on its face and what its rank will be. “If I have an [elliptic curve] and I tweak the coefficients, the rank changes drastically,” Park said. “You could change a coefficient by one and the rank could jump by a million. No one knows how ranks behave.”

This lack of a general theory forced mathematicians to fall back on the little evidence they had in order to guess whether ranks are bounded or not. “The viewpoint seemed to be that ranks are unbounded because people kept finding bigger and bigger ranks,” Granville said. The current record-holder is an elliptic curve with rank 28, which was discovered in 2006 by Noam Elkies, a mathematician at Harvard University.

But then this new model came along and said that, actually, it’s almost certain that the trail stops, and just up ahead.

A Surprise at 21

Scientists use models to study phenomena that are too complicated or forbidding to study directly. By creating a black hole analogue in a laboratory, you might be able to learn something about how real black holes behave without having to skirt an event horizon.

Mathematicians do the same thing. A good example comes from the study of prime numbers. Mathematicians would like to know the answer to the twin primes conjecture, which asks whether there are infinitely many pairs of primes whose difference is two (like 3 and 5, and 11 and 13). A complete answer is beyond their ken, but they’ve created models that predict how often twin primes should appear — and, indeed, the answer seems to be that they occur infinitely frequently.

The new model does not actually look at elliptic curves themselves. Instead, it examines a mathematical object called the kernel of a matrix. Kernels are to elliptic curves as mice are to humans — they’re not the same, but they’re easier to study and hopefully close enough that it’s reasonable to draw conclusions about one based on experiments on the other. In particular, kernels have their own version of rank. By looking at the distribution of ranks of kernels — how many kernels have rank 1, how many have rank 2, and so on — the four mathematicians hoped to get a feel for the distribution of ranks of elliptic curves. In essence, they were betting that the distribution of ranks of kernels and of elliptic curves mirror each other.