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
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).