But legitimate proofs are rare.
In the 1970s, mathematicians showed that almost all Collatz sequences—the list of numbers you get as you repeat the process—eventually reach a number that’s smaller than where you started—weak evidence, but evidence nonetheless, that almost all Collatz sequences incline toward 1. From 1994 until Tao’s result this year, Ivan Korec held the record for showing just how much smaller these numbers get. Other results have similarly picked at the problem without coming close to addressing the core concern.
“We really don’t understand the Collatz question well at all, so there hasn’t been much significant work on it,” said Kannan Soundararajan, a mathematician at Stanford University who has worked on the conjecture.
The futility of these efforts has led many mathematicians to conclude that the conjecture is simply beyond the reach of current understanding—and that they’re better off spending their research time elsewhere.
“Collatz is a notoriously difficult problem—so much so that mathematicians tend to preface every discussion of it with a warning not to waste time working on it,” said Joshua Cooper of the University of South Carolina in an email.
An Unexpected Tip
Lagarias first became intrigued by the conjecture as a student at least 40 years ago. For decades he has served as the unofficial curator of all things Collatz. He’s amassed a library of papers related to the problem, and in 2010 he published some of them as a book titled The Ultimate Challenge: The 3x + 1 Problem.
“Now I know lots more about the problem, and I’d say it’s still impossible,” Lagarias said.
Tao doesn’t normally spend time on impossible problems. In 2006 he won the Fields Medal, math’s highest honor, and he is widely regarded as one of the top mathematicians of his generation. He’s used to solving problems, not chasing pipe dreams.
“It’s actually an occupational hazard when you’re a mathematician,” he said. “You could get obsessed with these big famous problems that are way beyond anyone’s ability to touch, and you can waste a lot of time.”
But Tao doesn’t entirely resist the great temptations of his field. Every year, he tries his luck for a day or two on one of math’s famous unsolved problems. Over the years, he’s made a few attempts at solving the Collatz conjecture, to no avail.
Then this past August an anonymous reader left a comment on Tao’s blog. The commenter suggested trying to solve the Collatz conjecture for “almost all” numbers, rather than trying to solve it completely.
“I didn’t reply, but it did get me thinking about the problem again,” Tao said.
And what he realized was that the Collatz conjecture was similar, in a way, to the types of equations—called partial differential equations—that have featured in some of the most significant results of his career.
Inputs and Outputs
Partial differential equations, or PDEs, can be used to model many of the most fundamental physical processes in the universe, like the evolution of a fluid or the ripple of gravity through space-time. They arise in situations where the future position of a system—like the state of a pond five seconds after you’ve thrown a rock into it—depends on contributions from two or more factors, like the water’s viscosity and velocity.
Complicated PDEs wouldn’t seem to have much to do with a simple question about arithmetic like the Collatz conjecture.
But Tao realized there was something similar about them. With a PDE, you plug in some values, get other values out, and repeat the process—all to understand that future state of the system. For any given PDE, mathematicians want to know if some starting values eventually lead to infinite values as an output or whether an equation always yields finite values, regardless of the values you start with.
For Tao, this goal had the same flavor as investigating whether you always eventually get the same number (1) from the Collatz process no matter what number you feed in. As a result, he recognized that techniques for studying PDEs could apply to the Collatz conjecture.