Collatz proven for primes = proven for all integers? I recently saw a video stating that if the Collatz conjecture was proven for prime numbers, it was proven for all numbers.
That's the first time I see this and I can't find any references with the classical JFGI.
It says something like this ($a$, $b$, $n$ are odd integers):
$\sigma_p(n)$ is the Collatz trajectory ($p$ steps applied to $n$)
and for a fixed function $f(b)$ not provided you have:
$\sigma_p(an+b)=a\cdot\sigma_p(n)+f^p(b)$
and specially for odd prime factorization:
$\sigma_p(an)=a\cdot\sigma_p(n)+f^p(0)$
Apparently, it can be shown that if $\sigma(n)<n$ than $\sigma(an)<an$ and it suffice to show for all prime $n$ since a finite stopping time for primes implies a finite stopping time for all its composite
I tried on some numbers (e.g., $85=5\cdot 17$) but couldn't figure it out.

Is there a paper on this subject? The video seems to talk about a $15$ page proof.


To prove this, you’d likely show “for any n >= 2, the collatz sequence leads to an integer < n, or to a prime.”
This is most likely true, but very hard to prove, because the sequence is quite chaotic. That’s the same as the original problem, which is most likely true, but very hard to prove, because the sequence is quite chaotic.
If someone had a proof for this conjecture that could be very insightful. I’d be surprised.
