The most famous trajectory of $3x+1$ problem

Question

I think that the most famous and beautiful trajectory of the $3x+1$ problem is without doubt that starting from $n=27$ and having a maximum at $9232$.

The thing that I find very beautiful is that:

$$19\cdot 3^3=513\equiv 1\pmod {2^k}$$

And

$$\frac{9232}{2}=19\cdot 3^5-1$$

Is it chance or there is a Deep connection with the prime $19$ and the order of $19\cdot 3^s\pmod {2^k}$?

Are there other trajectories with similar features?

What is meant with "similar features" ? Trajectories where $19$ plays a role like in $27$ ? — Peter, Oct 04 '23 at 16:35
@Peter not only 19 but another prime p such that $p\cdot 3^j\equiv 1\pmod {2^k}$ and $p\cdot 3^{j+i}-1=\frac{M(n)}{2}$ — , Oct 04 '23 at 16:41
In the above example we would however have "$j+2$" instead of "$j+1$". — Peter, Oct 04 '23 at 16:43
My favourite trajectories are the trajectories of the numbers $-1$ and $0$. — it's a hire car baby, Oct 11 '23 at 12:01

score 5 · Answer 1 · answered Oct 04 '23 at 17:12

That's pretty common in the Collatz trajectories. Take any odd number and write it as $a\cdot2^n-1$, it will always climb up to $a\cdot 3^n-1$ in exactly $n$ steps of the function $f(x)=\frac{3x+1}{2}$. From there it will be divisible by $2^k$ with $k>0$. So just choose $a$ prime and you will find plenty of them with various value of $k$. $$a\cdot3^n\equiv 1\bmod(2^k)$$ That's called a "1-cycle"

The most famous trajectory of $3x+1$ problem

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