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I'm trying to solve the Collatz conjecture, and am having some trouble designing a function that divides a number by two until it's odd.

Here is what I've thought of.

We know $a\mod b = \arctan(\tan(\frac{a\pi}{b}-\frac{\pi}2))\frac{b}\pi + \frac{b}2 $

Thus,

$$f(x) = \frac{x}2 + \frac{x}2(x \mod 2) = \frac{x}2 + \frac{x}2(arctan(\tan(\frac{x\pi}{2}-\frac{\pi}2))\frac{2}\pi + \frac{2}2) = \frac{x}2 + \frac{x}2(arctan(\tan(\frac{x\pi}{2}-\frac{\pi}2))\frac{2}\pi +1)$$

What would this do? This function would return half the number, if $ x \mod 2 = 0$, else, return the whole number, because it's already odd.

The thing is, if I want to make sure any number is odd, I'd have to iterate this infinite times. I know some functions are easily iterated, but can all functions be iterated in theory? I know it may be hard, but the whole conjecture is tough by itself, so I don't really care.

One extra question, notice

$$a\mod b = \arctan(\tan(\frac{a\pi}{b}-\frac{\pi}2))\frac{b}\pi + \frac{b}2 = (\frac{a\pi}{b}-\frac{\pi}2)\frac{b}\pi + \frac{b}2$$

Is this simplification correct?

EDIT: The simplification collapses to $ a \neq a \mod b$, so that's out of the question.

ChemiCalChems
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  • Why would you use trig functions to such an end? Just write $x$ in base two, and let your function remove all the zeros from the least significant end (shifting the bits towards the low end by an appropriate number of positions). – Jyrki Lahtonen Mar 18 '17 at 12:33
  • How would you write that in a functional form? I'm trying to get to a proof for the Collatz conjecture, not just check numbers at random. – ChemiCalChems Mar 18 '17 at 12:34
  • Something like: While (x AND 1)==0 do (x>>1); – Jyrki Lahtonen Mar 18 '17 at 12:37
  • Again, I'm not trying to write a computer program. I want a mathematical function to use in the proof for the conjecture. – ChemiCalChems Mar 18 '17 at 12:38
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    Then $f(x)=x/2^{\nu_2(x)}$, $\nu_2(x)$ is the mathematical function that calculates how many times $2$ is a factor of the integer $x$. – Jyrki Lahtonen Mar 18 '17 at 12:40
  • This has been discussed in another question I have already posted. The thing is, how does one generalize $\nu_2(x)$? Maybe that is the real question. – ChemiCalChems Mar 18 '17 at 12:42
  • @JyrkiLahtonen : I've supposed in my answer in the other question, that the OP's wish is to have some expression of $\nu_2()$ as *function composed by elementary continuous functions*. (I don't think that is possible, btw.) – Gottfried Helms Mar 18 '17 at 13:44
  • For instance an exponentialtower with base $1<b<e^{1/e}$ can be seen as iterable $b^{b^b{^{...b}}} $ and can even be evaluated for the infinite iterate to a finite value solved by $c^b =c \to b=c^{1/c}$ or be solved for $c$ in terms of $b$ and of the Lambert-W-function of $\ln(b)$ – Gottfried Helms Mar 18 '17 at 13:51
  • @JyrkiLahtonen I think the answer to your "why?" question is to turn a discrete function into something with some nice properties such as a continuously differentiable function that takes on the appropriate values at integers - akin to the Gamma function for factorials so that calculus can be brought into play. – Robert Frost Mar 23 '17 at 16:22
  • @RobertFrost: Possibly :-) I haven't investigated the matter, but I am not sure the series I gave in my answer is differentiable. It is continuous as the sum of a uniformly converging series of continuous functions. But I don't think the series formed by termwise derivatives converges uniformly. At least I'm not sure it does. Therefore there is no guarantee about the sum being differentiable. – Jyrki Lahtonen Mar 23 '17 at 16:47

2 Answers2

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If you insist on using trig functions you can do the following. Consider the sequence of functions $$ \begin{aligned} f_0(x)&=1,\\ f_1(x)&=\frac12(1+\cos\pi x),\\ f_2(x)&=\frac14(1+\cos\frac{\pi x}2+\cos\pi x+\cos\frac{3\pi x}2),\\ \vdots&\qquad\\ f_k(x)&=\frac1{2^k}\sum_{j=0}^{2^k-1}\cos\frac{j\pi x}{2^{k-1}},\\ \vdots&\qquad \end{aligned} $$ At integer points they have the properties:

  • $f_0(n)=1$ for all $n\in\Bbb{N}$,
  • $f_1(n)=1$ when $2\mid n$ and $f_1(n)=0$ at other integer points,
  • $f_2(n)=1$ when $4\mid n$ and $f_2(n)=0$ at other integer points,
  • $f_k(n)=1$ when $2^k\mid n$ and $f_k(n)=0$ at other integer points.

Therefore the series $$ F(n):=f_0(n)-\frac12f_1(n)-\frac14f_2(n)-\cdots=1-\sum_{k=1}^\infty 2^{-k}f_k(n) $$ has the value $F(n)=2^{-\nu_2(n)}$ at any integer $n$.

Consequently $$ x F(x) $$ divides an integer $x$ by all the powers of two that still produce an integer.

Unless I made a mistake the series $F(x)$ converges uniformly everywhere, so it is a continuous function. Here's a plot of the sum of the first five terms, call it $S_4(x)$, in the interval $x\in[0,16]$.

enter image description here

Here's the corresponding plot of $x S_4(x)$.

enter image description here

I doubt you can approach Collatz in this way, but who am I to forbid anyone to think outside the box :)

Jyrki Lahtonen
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  • So if I understand, the function gives the odd number that arises after dividing by two as many times as possible? You sure are a genius. I admire you deeply. In case this thinking-out-of-the-box arrives somewhere, I will cite you everywhere, because this is the cornerstone I was looking for. – ChemiCalChems Mar 18 '17 at 17:35
  • Now I'll only have to deal with infinite summations, but that isn't really that big of a problem. – ChemiCalChems Mar 18 '17 at 17:59
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Let $f(x) = b^x $ with $b=\sqrt{2}$ then $f(1)$,$f(f(1))$, ..., $f°^h(1)$ are all well defined and computable and converging to $c=2$.
We can even use the analytial result that the limit of infinite iteration is equal to $2$ and can therefor derive from that any further (appropriate) corrolary, lemma or theorem in all generality (well, for bases $b$ with appropriate values) ...

Another example: let $g(x) = x/2 $ then we can iterate $g(1)$,$g(g(1))$ ... $g°^h(1)$ and even can say an analytical result for the sum of all iterates : this is the geometric series and evaluates to $1+ g(1) + g°^2(1) + g°^3(1)... = 2$

Gottfried Helms
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  • I have been quite interested in iterated functions for some time, and I realize functions as trivial as these are computable. The problem is with the function above. I'm sure I'd take ages, but it must be possible for any function, right? – ChemiCalChems Mar 18 '17 at 14:15
  • @ChemiCalChems : Well, I'm not a professional mathematician and I cannot say, whether your intended function exists or not (I doubt that it exists) but your question was then insofar a bit unspecific: I'd understood you asked whether there is any function with the analytical property when iterated, so ... – Gottfried Helms Mar 18 '17 at 14:20
  • When I said any, I meant all functions. There has to be a simpler way to solve the conjecture. Maybe it really has to do with finding $\nu_2(x)$ in terms of elemental functions. – ChemiCalChems Mar 18 '17 at 14:21
  • @ChemiCalChems : yes, I think that would be the best expression for other mathematicians to understand the real question... – Gottfried Helms Mar 18 '17 at 14:22
  • Edited the question. – ChemiCalChems Mar 18 '17 at 14:25