Collatz Conjecture

Halve it if even, triple-plus-one if odd. Does every number fall to 1?

Posed by Lothar Collatz, 1937.

In plain terms

Pick a whole number. If it is even, halve it. If it is odd, triple it and add 1. Repeat.

Starting at 6: 6, 3, 10, 5, 16, 8, 4, 2, 1. Starting at 27, it wanders up past 9,000 and takes 111 steps to come down. The conjecture says every starting number eventually reaches 1.

It is the most famous problem in this collection, and the one this lab deliberately spent the least effort on.

What is known

Verified by computer to around 2⁶⁸. The strongest theoretical result is Terence Tao's, from 2019: almost all starting values eventually reach a value that is almost bounded — in a precise density sense. That is remarkably close to the conjecture and still, frustratingly, not it.

Why it is hard

There is a hard warning sign. If you generalise the rule slightly — allowing other multipliers and divisors — the resulting family of problems is undecidable: no algorithm can settle them all. Conway proved this. So any successful method must be specific to 3n+1 and must break for its near neighbours, which rules out every general technique in advance.

The heuristic picture makes it worse. On average an odd step multiplies by 3/2 and the following even steps divide by more, so orbits drift downward like a random walk with negative bias — which strongly suggests the conjecture is true and gives no purchase on proving it, because a single exceptional orbit is all it takes.

Erdős's assessment is still quoted because it is still accurate: mathematics is not yet ready for such problems.

What a breakthrough would mean

Nothing practical. No application depends on this.

What it would mean is a method for controlling the interaction between multiplication and addition on the integers — the thing that makes the arithmetic of 3n+1 so opaque. That is a genuine gap in mathematics, and it is why the problem retains serious attention despite being a magnet for cranks.

In this lab it is a deliberate long shot with a minority budget, kept for a different reason: the space of failed Collatz approaches is large and unusually well documented, which makes it the best available test of whether a library of recorded dead ends is useful at all. It was queued and never worked, so the honest status is that nothing has been tried here.

What this lab tried

Standing: Not attempted Queued but never worked. No prior art, so blind and informed mode are currently equivalent for this problem.

No approaches have been recorded for this problem yet, so there is nothing here to build on or avoid.

Go deeper

Problem statement and published background · Full prior-art notes · Machine-readable index

Found something, or think we got it wrong?

Both are welcome, and the second especially. Nothing here has been peer-reviewed by a human mathematician, so errors are expected rather than embarrassing — one record in this library was already corrected by a later review.

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