# Are there any partial proofs for the collatz conjecture? [duplicate]

For my math internal assessment, I'm looking at the Collatz Conjecture and different ways to try and solve it.

I wanted to see if there were any possible partial solutions for it by which I mean if we take a specific type of number the conjecture would be true for all of those numbers.

For example, the collatz conjecture is true for all m where m=2^n since the powers of two will keep being even until it reaches one. However, I wanted a more complex proof for another type of numbers(Say, for all powers of 3 or for all prime numbers if it were possible) so I wanted to know if there are any such partial proofs you could think of

## marked as duplicate by user99914, Robert Soupe, Ethan Bolker, rtybase, José Carlos SantosJul 1 '18 at 21:36

• Please specify which sets are allowed! If it is the set of odd numbers $n$, such that $3n+1$ is a power of $2$, then the Collatz-conjecture is trivial to prove. – Peter Jun 4 '18 at 10:43
• There really isn't a specific type of sets are allowed which means the one you've given is also viable I just wanted to see the different kind of sets that did exist for which the conjecture could be proven but ideally I would like non-trivial proofs with some mathematics that is worthy of investigation and is of some high level since the one you've given is quite self explanatory. THank you – Samik Maini Jun 4 '18 at 11:44
• see my comment here math.stackexchange.com/questions/2716155/… – Collag3n Jun 4 '18 at 17:23

The Collatz conjecture is proven for:

$$2^n$$

$$2^n\cdot x$$ for all integers $$x$$ for which it is proven.

Every positive integer up to some really large number like say $$2^{50}$$

Every number of the form $$\frac {4^{n}-1}3$$; i.e. $$0,1,5,21,85,341,\ldots$$

Every number of the form $$4^n\cdot x+\frac{4^n-1}3$$ for all odd $$x$$ for which it is proven, i.e. if $$3$$ converges then $$3,13,53,213,\ldots$$ converges.

Every $$4x+1$$, for all odd $$x$$ that converge.

Every linear combination of the Lucas sequences $$U_n(5,4)$$ and $$V_n(5,4)$$, for which the first element of the sequence is a convergent odd integer.

$$\frac{y-1}{3}$$ converges for every $$y=64^n\cdot x+\frac{64^n-1}{3}$$, for every $$\frac{x-1}{3}$$ known to converge.

A number of these are largely equivalent and proving any of these is a relatively simple exercise and should keep you going for a while (the $$\{x<2^{50}\}\to1$$ exercise requires a computer and some time).