Pick any positive whole number. If it is odd, multiply by three and add one; if it is even, halve it. Repeat. Collatz conjectured in the 1930s that every starting value eventually reaches the number 1, and then loops. of examples have been tested without an exception, and nobody can prove it.
Because it is simple enough to attempt and hard enough to swallow a career. Quanta quotes an expert calling it a really dangerous problem that people become obsessed with, and reports that the internet is awash in unfounded . In 2019 Terence Tao proved the conjecture is almost true for numbers.
The rules are a . Pick a number. If it is odd, multiply it by three and add one. If it is even, divide it by two. Apply the same rules to the result, and keep going.
Intuition suggests where you start should decide where you end: some numbers spiralling down, others marching off to infinity. Collatz predicted otherwise, that every positive whole number eventually reaches 1. After that the rules confine it to a loop of three values, round and round.
The evidence is enormous and the proof is absent. Mathematicians have tested quintillions of examples without a single exception, and the internet is awash in unfounded amateur proofs. Lagarias, an expert on the problem, calls it a really dangerous problem that people become obsessed with.
In September 2019 Terence Tao posted the most significant result in decades: the conjecture is almost true for almost all numbers. He said he had not expected to solve the problem completely, and that what he did was more than he expected.
wat. Three times plus one, or halve it, and nobody can prove you always land on 1.
wat wat
A rule a child can follow can still outrun every method mathematics has, which is a fact about mathematics rather than about the rule.
How do we know?
Quanta Magazine reports Tao’s September 2019 result, quotes him and Jeffrey Lagarias on the problem, and sets out the rules and the state of testing.
What is still unknown
This is magazine reporting on a result rather than the paper, and Tao’s theorem is not a proof of the conjecture.
Learn something?
Where to next
Can one shape tile a floor that never repeats?open problems
How do you prove the primes never run out?number theory
What makes 1729 an interesting number?number theory