I have been getting into the Collatz Conjecture and I have noticed something interesting. Consecutive numbers often take the same amount of numbers to get to 1. For example, $84$ goes to $42$ to $21$ to $64$ to $32$ to $16$ to $8$ to $4$ to $2$ to $1$. That's ten numbers. $85$ goes to $256$ to $128$ to $64$ to $32$ to $16$ to $8$ to $4$ to $2$ to $1$. That's also ten numbers. There are many more examples, like $60$ and $61, 76$ and $77,$ and $92$ and $93.$ As the numbers get larger, the number of consecutive integers that take the same amount of numbers to get to one grows. $386, 387, 388, 389, 370,$ and $371$ all take $121$ numbers to get back to $1$! I have thought a lot about this, but I can't figure out why this pattern works. I also cannot find a pattern of when this pattern occurs. It seems to be random, but I might be wrong. Can someone help with these questions?
Why does this pattern with consecutive numbers in the Collatz Conjecture work
collatz conjecturenumber theory
Related Solutions
I will answer your question: I bet you won't like my answer but it is answer so I hope you keep that in mind.
Your question was: "I would like to know, if I could try harder at this pattern (I am stuck), if it could lead to a proof. Or it would just be other thing to waste my time here."
When I was a younger math nerd I attempted to take the P vs NP problem very seriously and after burning out a few times a mathematician and mentor of mine told me that "This is not to be done." He wanted me to focus on acquiring some mathematical tools and some experience that would help me become a more competent math nerd and wanted to give me a warning about diving into exceptionally difficult mysteries without sufficient experience.
Questions like the Collatz conjecture are good for getting mathematicians to wake up but not a good place to spend your time.
I mean that this is the type of thing that mathematicians as a culture should be on the same page about: when you see a young and-up-and-coming mathematician (Or really any non pro) making serious efforts towards a problem that has been open for a century we should dissuade them from getting bogged down in these mental traps.
This is a great puzzle. And it's very accessible which makes it a nice tool to get students excited about open mathematics. But also very very impressive mathematicians have said things along the lines of "we're not quite ready to solve such mathematical mysteries. We don't have all the tools yet."
The reasons you should not spend more time on this puzzle.
1) With probablity approaching 1, better mathematicians have already attempted a route similar to the one above. Which (and without any insult intended here) doesn't look particularly novel.
2) The problem isn't of particular interest. We have more important work to be doing: work that we can actually make real progress on.
3) It's probably a dead end. I mean... thus far all the numbers we have tested head to one... and all of the attempts we have made(for a century) have led to either undecidable generalizations or just a flat failure to say anything interesting.
You should enter the mathematical community with burning questions like this one... and then when you seen the human effort put toward this question and have read Conway's book on the 3x+1 problem and checked out Lagarias and whoever else you need to convince yourself that very impressive minds have spent considerable energy on this...
You should put down these puzzles and help make genuine mathematical progress. It will be better for you and the mathematical community at large.
There is a subtle issue with your induction argument: you are assuming that the Collatz conjecture holds for all integers $\leq n$, and then want to prove it holds for $n+1$ (strong induction). So far, so good.
You then prove that for some cases ($n+1$ even, or of the form $4k+1$) that the Collatz conjecture holds by the inductive hypothesis. Fine.
You then try to argue that for some numbers of the form $4k+3$, you eventually hit a number of the form $4k+1$, so that the Collatz conjecture holds... not so fast. You haven't proven that the Collatz conjecture holds for all integers of the form $4k+1$. You've proven it's true for $n+1$, if $n+1$ happens to be of that form, and you've assumed it's true for all numbers of that form $\leq n$ (by the inductive hypothesis) but you haven't shown that Collatz holds for numbers of the form $4k+1$ that are larger than $n+1$.
Best Answer
There are indeed patterns of this form.
If, say, you start with a number of the form $8n+4$ the chain begins
$$8n+4\mapsto 4n+2\mapsto 2n+1\mapsto 6n+4$$
While if you add $1$ to get $8n+5$ you get $$8n+5\mapsto 24n+16\mapsto 12n+8\mapsto 6n+4$$
Thus the consecutive numbers $8n+4,8n+5$ always have the same Collatz length. That explains your pairs $(84,85)$, $(60,61)$, $(76,77)$, $(92,93)$. I expect there are other patterns as well.