Can MathGPT solve the Collatz conjecture (3n + 1 problem)?

No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

UnsolvedNumber theory

Collatz conjecture (3n + 1 problem)

Halve if even, triple and add one if odd: every start eventually reaches 1?

Exact statement

f(n)=\begin{cases}n/2,& n\ \text{even}\\ 3n+1,& n\ \text{odd}\end{cases}\qquad \forall n\ge1\ \exists k:\ f^{k}(n)=1

For every positive integer n, repeatedly applying f eventually reaches 1.

Status

Open. Posed by Lothar Collatz in 1937; checked by computer for every start below 2⁶⁸.

Posed: 1937, Lothar Collatz.

In plain words

Start with any positive whole number. If it is even, halve it; if it is odd, multiply by 3 and add 1. From 6 you get 6, 3, 10, 5, 16, 8, 4, 2, 1. The conjecture says you always end at 1 (then cycle 1, 4, 2, 1).

The rule is simple but the sequences behave erratically: 27 takes 111 steps and climbs to 9232. Erdős reportedly said mathematics 'is not yet ready for such problems'.

Key results so far

1972: Conway shows a natural generalisation of Collatz-type problems is algorithmically undecidable.
1976: Terras: almost all n (in density) eventually drop below their starting value.
2019: Tao: almost all orbits (in logarithmic density) attain almost bounded values.
2021: Barina verifies the conjecture for every starting value below 2⁶⁸.

What our solver can and cannot do

No computer program or AI can solve the Collatz conjecture (3n + 1 problem) by being asked. If you paste it into the MathGPT solver, you get this reference information, labelled as such, never an invented proof. The solver can still compute related special cases and examples. Open the solver.

Frequently asked questions

What does the Collatz conjecture (3n + 1 problem) say?: For every positive integer n, repeatedly applying f eventually reaches 1.
Has the Collatz conjecture (3n + 1 problem) been solved?: Open. Posed by Lothar Collatz in 1937; checked by computer for every start below 2⁶⁸.
Can MathGPT solve the Collatz conjecture (3n + 1 problem)?: No. No one has a proof, so no solver or AI can give one. MathGPT recognises the problem and shows the exact statement, its status and the established facts, labelled as reference information.

References

J. Lagarias (ed.), The Ultimate Challenge: The 3x+1 Problem, AMS (2010).
T. Tao, arXiv:1909.03562 (2019).
D. Barina, 'Convergence verification of the Collatz problem', J. Supercomputing (2021).

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