Investigating natural numbers in relations with prime numbers If we look at any initial condition or any configuration in a collatz-sequence, do we find patterns to primes? I just found out the first 16 possible configurations adds up to the primes, but I need to check if this is true for numbers up to a hundred or a thousand to be sure this is true.
My research-notes have gone over 30 pages so far. I guess thats not a big deal for those who have written dozens of papers. I am probably the only one who understands my own work, and need to practise more on writing it in a formal matter. So there's no point for me to share my notes so far. Since I am looking at the Collatz-problem from a simple perspective, I just wonder, are there any simple basic research in this area, or does it involve complicated equations?
To reiterate my question, it goes like this:
Is there any patterns to primes in relations to an initial condition or any configuration in a Collatz sequence?
And if someone have an answer, please answer in simple terms, I would appreciate that.
Thanks.
 A: Prime number "patterns" in the Collatz Conjecture are most likely coincidence, unless the configuration requires finding primes in the first place. Since a lot of Collatz research focuses on odd numbers, one is more likely to come across prime numbers simply by eliminating the even numbers. When you introduce the "short cut" map of the Conjecture where if $x$ is odd, then $x$ = $(3x+1)/2$, this reduces the amount of time it takes to get another odd number, thus increasing the chance of finding more prime numbers.
There is research on the connection between the Collatz Conjecture and the Mersenne primes. More can be found on that here. As for the Jacobsthal-Lucas numbers, I believe it is also a coincidence since they are similar to the Mersenne primes. 
As for other related Collatz pattern formulas, Gottfried Helms explains it better here.
Aside from that, I can only speak from experience that I have unsuccessfully found a connection between prime numbers and the Collatz Conjecture.
A: One of two operations in Collatz Conjecture is:


*

*if the number is $odd$, triple it and add $1$.


One of the polynomial form for all prime numbers is:


*

*if the number is $odd$, triple it and add $2$ or $4$.

