The Collatz Conjecture: Mathematics' Simplest

Pick any positive integer. If it's even, halve it. If it's odd, multiply by three and add one. Repeat until you can't anymore.

Every number anyone has ever tried eventually lands on 1. Every single one, across 89 years of trying, including a 2025 supercomputer sweep by researcher David Bařina that verified every positive integer up to 2⁷¹—a 22-digit number, roughly a sextillion cases. And yet nobody can prove this always happens. That gap, between every case we've ever tested and every case that exists, is the entire problem.

This is the Collatz conjecture, and a new documentary from STEM in Motion by Gaurav does something most popular treatments of this problem don't: it takes the negative numbers seriously. That decision reframes the whole puzzle.

The hailstone problem

Before you can appreciate why the conjecture resists proof, you need to feel its texture. The number 27 is a decent place to start. It's odd, so you triple it and add one: 82. Halve that: 41. Odd again, so triple and add one: 124. From there, the sequence rockets to 9,232—more than 300 times the starting value—before eventually grinding back down to 1 after 111 steps. The number 26 takes 10 steps. The number 28 takes 18. Sitting between them, 27 takes over a hundred.

Mathematicians call these hailstone numbers, and the metaphor earns its keep. A hailstone doesn't fall cleanly to earth—it gets thrown back into clouds by updrafts, cycling through the storm until it's finally too heavy to stay aloft. There's no formula that looks at a number and predicts how high it'll climb or how many steps it'll take. You can only run it and watch.

This unpredictability is partly what kept the problem circulating underground for decades. Lothar Collatz noticed the pattern in 1937 and never published it—just passed it around. It re-emerged at different institutions under different names: Kakutani's problem at Yale, Ulam's problem at Los Alamos. Four mathematicians, one rule, zero papers. When the mathematical community eventually realized they were all staring at the same thing, they set out to settle it. They still haven't.

Why the computers aren't enough

The computational effort is genuinely staggering. Bařina's 2025 verification used thousands of processors and confirmed the conjecture holds for every positive integer up to 2⁷¹. In experimental science, that kind of empirical saturation would close the case. In mathematics, it doesn't even constitute evidence in the technical sense.

The reason lives in a cautionary tale from 1919, when George Pólya proposed a conjecture about prime factors that held for every number tested—through the 1920s, 1930s, and 1940s. Its first counterexample sits at just over 900 million. A pattern that held for 900 million consecutive cases before breaking. Against an infinite number line, even 2⁷¹ is a rounding error.

"Mathematics has never accepted 'we checked a lot of them,'" the video notes, "because checking was never the path."

So if checking can't close it, what about reasoning? Here's where the story gets structurally interesting.

The downhill staircase—and its blind spot

There's a genuine mathematical argument for why the Collatz conjecture should be true. When you triple an odd number and add one, the result is always even—which means every upward step is immediately followed by at least one halving. When you average out how many halvings typically follow each tripling, each odd number in the sequence ends up being roughly three-quarters the size of the odd number before it. A steady contraction. In logarithmic terms, the downward steps beat the upward steps by about 0.3 of a unit per step on average.

Picture a drunk staggering down a sloped street. He's not walking in a straight line. But the ground tilts downhill, and eventually, he gets to the bottom. The slope does the work; the weaving doesn't matter.

This isn't just hand-waving. When you swap 3n+1 for 5n+1—multiplying odd numbers by five instead of three—the same statistical logic predicts the staircase should tilt upward, meaning most numbers should escape to infinity. And that's exactly what happens. Of the first 199 starting values under the 5n+1 rule, only 33 ever reach 1. The seven, 1,900 steps in, has ballooned to a 61-digit number and is still climbing.

The staircase argument predicted the behavior of a rule nobody had previously tested. It was correct. So why isn't the Collatz case closed?

Because of the negative numbers.

What zero doesn't explain

The Collatz rules contain no restriction to positive integers. The arithmetic works on negatives just as cleanly. Run it on −5: triple and add one gives −14, halve to −7, triple and add one gives −20, halve twice—and you're back at −5. A closed loop. Start at −17 and you land in a different loop entirely, 18 numbers long, stretching down to −272 before cycling back. Start at −1 and you get the tiny loop −1 → −2 → −1.

Three distinct loops. On the negative side of the number line, the Collatz conjecture isn't unproven—it's false. It has been since 1937.

Now here's the part that should bother you: the downhill staircase argument applies identically to negative numbers. Tripling a negative odd number and adding one still yields an even number. The contraction ratio is still three-quarters. The slope is still 0.3 units downhill per step. Every single statistical reason to believe the conjecture on the positive side is a reason that is already demonstrably wrong on the negative side.

The staircase argument is blind to loops. A looping sequence doesn't violate the average downward drift—it just cancels out exactly as it cycles. As the video puts it: "The problem is that statistics can only see the average drift. It is completely blind to cycles, and a cycle is one of the three things that can happen."

This is the actual obstacle. Not computational scale. Not mathematical machinery being too crude. The intuition that underpins our confidence in the conjecture is an intuition we can watch fail in real time, two steps to the left of zero.

The two ceilings

Researchers haven't been idle. Working backward from 1, you can build a tree of every number that eventually leads there—and the Collatz conjecture is simply the claim that this tree contains every positive integer, with no detached loops floating elsewhere. Progress on ruling out those loops has been methodical: R. P. Steiner eliminated single-peak loops in 1977; by 2023, Christian Hercher had pushed the exclusion zone to 91 peaks. A rogue loop, if it exists, must have at least 92 peaks, consist entirely of numbers larger than 2⁷¹, and span at least 355 billion steps.

We know a lot about a thing that may not exist.

Two deeper obstacles explain why that "may not" feels so hard to resolve. The first is John Conway's 1972 discovery that systems like Collatz—rules that branch based on a number's remainder—can simulate any computation. This means asking whether all Collatz orbits reach 1 is structurally equivalent to asking whether an arbitrary computer program halts. And Alan Turing proved the halting problem is undecidable: no general algorithm can answer it for all cases. Collatz itself might be provable—it's one specific case in a family that contains unanswerable questions—but there's no way to know from the outside which kind it is. You could spend a career on a proof that doesn't exist.

The second ceiling is Terence Tao's 2019 result, the most significant advance in decades. Tao proved that for almost every starting number, the sequence will eventually drop below any threshold you name, no matter how slowly that threshold grows. This is vastly stronger than earlier results, and it's the closest mathematics has come to the finish line. But "almost every" has a precise mathematical meaning that tolerates infinitely many exceptions—and Tao has been explicit that his methods can't close that remaining gap. As the video summarizes his position: the remaining distance "is of a kind that his methods, and perhaps any current methods, simply cannot close."

Paul Erdős, not a mathematician given to excessive modesty, reportedly warned that "mathematics is not yet ready for such problems." Jeffrey Lagarias, who has spent decades on the conjecture, called it "completely out of reach of present-day mathematics." A 120 million yen bounty offered by a Japanese company in 2021 remains unclaimed.

What's genuinely strange about the Collatz conjecture isn't that it's hard. Mathematics has hard problems. What's strange is that the rule is two lines long, the failure case is visible with pen and paper three steps to the left of zero, and the best explanation we have for why it works on the right is an explanation we can watch not work on the left.

Nobody knows what changes at zero. That's not a rhetorical closer—it's the actual state of the field.

By Amelia Nwofor, Science Desk Editor