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.


I think every 5-rough integer is preceded by some prime number - I don't know a proof of this but the result claimed in the question does follows from it, if true. An argument in favour of this goes: If you take any 5-rough composite number and think how much freedom you have to find a prime that precedes it, you can pick from infinitely many Lucas Sequences.
These Lucas sequences contain no prime only under very limited circimstances, so every number is almost certainly preceded by some prime, which is precisely the same statement you're asking about.
You can see some rare cases in which whole classes of predecessors of some number are prime-free here: Does the sequence $x_0=8$ , $x_{n+1}=4x_n+1$ contain a prime? and in the questions linked to that.  None of these comes close to showing some infinite sequence of composite numbers.
