math discussion1022 - 2022-02-09

23:31

Q: The Collatz function (with jumps) is a monoid homomorphism, can we continue this along the iteration sequence?

The set of all $x\in \Bbb{Z}$ such that $\dfrac{3x + 1}{2^z} = 1$ for some $z \geq 0$ is closed under $x\cdot y := 3xy + x + y$. In fact this gives the set of iteration degree 1 Collatz solutions the structure of a monoid, with $0$ as the identity. For proof, just multiply out $1 = \dfrac{3x + ...

What's the first part that doesn't jive with you?

geocalc33

what does "with jumps" mean

PenAndPaperMathematics

Well you know how you only need to prove Collatz for odd numbers

since $x$ even means $x = 2^i y$ for some maximal $i$?

I'm simply dividing all the $2^i$'s out in one step. Collatz conjecture for this is equivalent to Collatz conjecture for the original function.

Because any $f^i(x) = 2^k$ for some $k \geq 1$ automatically means $x$ is a solution to the conjecture becayse $2^k / 2 = 2^{k-1}, 2^{k-1} / 2 = 2^{k-2}, \dots, = 1$.

geocalc33

okay

PenAndPaperMathematics

So it's like the submonoid $M = \{2^k : k \geq 0\} \leqslant (\Bbb{N}, \cdot)$ is a nice automatic highway down to $1$. Any time $f^i(x)$ has value in that monoid, i.e $f^k(x) \in M$, the proof is done for case $x$.

But you have to prove an infinite number of these cases!

So for example $f(3) = 5, f(5) = 1$. Since $f^2(3) = f(5) = \dfrac{3\cdot 5 + 1}{2^4} = 1$.

The original Collatz will maximally divide out $2$ just once, I'm saying go ahead and divide out all the $2$'s since the two statements are equivalent.

Thus I work with just the odd numbers.

geocalc33

23:49

got it

@PenAndPaperMathematics

PenAndPaperMathematics

Yes

geocalc33

say a vector space $\zeta$ has binary operations $\times$ and exponentiation

$\zeta = w_1^{\mathbb Z} \odot w_2^{\mathbb Z},$ where $w_1=(a,1)$ and $w_2=(1,b)$ for some $a,b>0; a,b\neq 1,$ and $(x_1,y_1)\odot(x_2,y_2)=(x_1 x_2, y_1 y_2)$ and $w^{\mathbb Z} = \{ w^n \mid n \in \mathbb{Z} \}$?

$K= w_1^{\mathbb Z} \oplus w_2^{\mathbb Z}$

PenAndPaperMathematics

K, supposed

geocalc33

what then would $K$ be?

PenAndPaperMathematics

Looks like the direct sum of your vector spaces

Write it using usual notation for better understanding

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