I found a pattern of repeating primes by using the odd/evenness of the number at each step of the Collatz conjecture procedure to get a new integer for every positive integer, which forms a sequence of integers (call it $C$).
For example, the Collatz steps of $6$ are
$$ 6\to3\to10\to5\to16\to8\to4\to2\to1 $$
We take these steps (excluding $1$) to obtain $(6,3,10,5,16,8,4,2)$. Then, take each element $a_i$ of the obtained tuple and replace it with $(a_i+1)\pmod{2}$. Then, reverse the elements to get $(1, 1, 1, 1, 0, 1, 0, 1)$. We interpret the result in binary to get $$c(6)=11110101_2=245$$
That is, we have $C=(c(1),c(2),c(3),\dots)$ if we do this for every $n\in\mathbb N$.
After taking the absolute differences of consecutive elements of $C$, we get $C'$. I noticed there were primes among them. Looking only at the prime numbers in $C'$, (call those $P$), if I took only elements of $P$ less than some $m$, it looks like the sequence repeats indefinitely, for any $m$ chosen. Call that last sequence $C_m$.
I'm glossing over a few implementation details in the interest of keeping this short.
The conjecture is: for all $n$ does $C_n$ repeat?
Is there a name for this conjecture and if so, what is it?
Here are some of the sequences:
$C' = 1, 121, 119, 27, 215, 63093, 63331, 506703, 506649, 15773, 15343, 3, 126183, 213, 126875, ...$
$P = 63331, 15773, 3, 3943, 13, 3, 31543, 53, 3, 13, 3, 23, 3, 103, 13, 3, 53, 4150522877, 3, 3413, 13, 3, 23, 3, 13, 3, 53, 1037618963, 1037630719, 3, 853, 13, 353, 3, ...$
$C_{13} = (3), (3), (3), ...$
$C_{23} = (3, 13, 3), (3, 13, 3), (3, 13, 3), ...$
$C_{53} = (3, 13, 3, 3, 13, 3, 23), (3, 13, 3, 3, 13, 3, 23), ...$
$C_{103} = (3, 13, 3, 53, 3, 13, 3, 23), (3, 13, 3, 53, 3, 13, 3, 23), ...$
(The parentheses above are included only to show repetition boundaries. Otherwise they can be ignored.)
As an aside I wrote some code in C using GMP to check for counter examples. It didn't find any and it checked quite a few. It doesn't mean much, but it's kinda interesting.
Edit: If you'd like to talk about this, please contact me at tyler at linkless dot org . It might be easier to talk about one on one and back port that here. Normally I wouldn't suggest that, but I'm kinda excited about this one.
I also realize I posted a related question: Is there a name for this "Collatz constant"? which might explain the procedure well enough.
Update:
So I was concerned about the above mention of 63,331 not repeating in one of the comments. I modified my algorithm to remove any prime that occurred only once in the entire sequence tested. I wrote code that took 10,000,000 collatz primes, and checked for repetitions per the modified algorithm. The preliminary search hasn't found a counter example.
I've checked for repetition with a maximum prime for the first smallest 256 collatz primes. Here are the results: https://samplesizeofone.com/external/stack_overflow/collatz-primes-stats.pdf
For any given run, all collatz primes above max prime are removed. The sequence length is the smallest first n elements that, when repeated, produce the rest of the tested sequence. Sequence repetitions is how many times the found sequence needs to be repeated to equal the tested sequence.
Here's a similar output. The interesting bit is the graph of the repeated sequence. Note, the numbers on the vertical axis are indexes, not the primes themselves. But they are, for all intents and purposes here, interchangeable. https://samplesizeofone.com/external/stack_overflow/collatz-repetitions-graph.pdf
Clearly failure to find a counter example for 256 cases in 10,000,000 collatz primes doesn't mean a ton, but there we go.
It looks like no one so far has a name for this, so for the purposes of this question, I'll assume the answer is no.