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For all $x \in \mathbb{N}$ and $y \in \mathbb{N}$,

$$ Q= \frac{2^x-3^x}{3^x-2^{x+y}}$$

the only time Q is a natural number and odd is when $(x,y)=(1,1)$.

I've been trying to solve this for a while as this equation came out of a simplified case of a problem I am working on but I don't know how to consider all the cases.

I have plotted this in Matlab before and I know that when $x \gg y$ it produces very large negative numbers and when $y \gg x$ the limit goes toward zero. But, there is a "line" when $x$ is slightly greater than $y$ that causes the values become weird and sporadic. The last thing I know is that it is easy to see that the only time Q is positive is when $ y > x \log_2 (3/2)$. What methods and/or ideas would help prove this? Thanks in advance.

Edit: I forgot to mention in the orignal post but I already know that the cases of $x=y$, $x \gg y$ and,$y \gg x$ back up this statement, I was asking about how I could go about proving the cases where x and y aren't drastically different, thus causing positive numbers close to natural numbers.

Gottfried Helms
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Kiyro6
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  • In my studies I use $N$ for $x$ and $S$ for $x+y$ (thus $B$ for $y$) and $N$ denotes the number of odd steps in the Collatz-iteration and $S$ the number of even steps. Then $S=\lceil N \cdot \log(3)/\log(2) \rceil$ to have $Q$ positive. When $N$ and $S$ are convergents of the continued fraction of $\log(3)/\log(2)$ then $Q$ is very near to integers, and one can assume, that this can be arbitrarily near to integers with increasing $N$ Compare for instance http://go.helms-net.de/math/collatz/2hochS_3hochN_V2.htm to see examples of such $N$ and $S$ – Gottfried Helms Dec 18 '17 at 10:14
  • This looks equivalent to the Collatz conjecture; a famous unsolved problem. It is likely you need to create some metric with measures the distance between your quotient and the number $1$. Also look at the 2-adic extension of the Collatz Conjecture mentioned on its Wikipedia page. – it's a hire car baby Dec 19 '17 at 15:41

1 Answers1

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In the discussion around the Collatzproblem we encounter the OP's formula in similar form: $$ \Large \begin{array} {} Q&= \frac{2^x-3^x}{3^x-2^{x+y}}\\ &=\frac{2^x-3^x-2^{x+y}+2^{x+y}}{3^x-2^{x+y}} \\ &=\frac{2^x-2^{x+y}}{3^x-2^{x+y}}-1 \\ &=2^x \frac{2^y-1}{2^{x+y} - 3^x}-1 \end{array} $$ and the conjecture: $$ x=y=1 \Leftarrow \Rightarrow Q \in \mathbb Z^+$$ The only proof that there is no other positive integer solution, that I know of, is the proof of Ray Steiner (using transcendence-theory) from 1978. Later this proof has been somewhat simplified (better bounds for the involved diophantine approximation have been made available) but there has not been found an elementary way to prove this nonexistence of a positive integer solution for $x,y \gt 1$ . (see an example for that proof at an older answer of mine in section "update 2" and also an extraction/ extension focused only at this style of proof at my homepage.)

Gottfried Helms
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  • I also found this equation while working on the Collatz Conjecture, I quickly looked at the link you posted so I can say for sure but I believe were approaching it differently so I find it fascinating we both derived this equation. Thank you. – Kiyro6 Dec 18 '17 at 19:29
  • @Kiyro6 - ah, well, you're welcome! Perhaps you like this: http://go.helms-net.de/math/collatz/Collatz061102.pdf where I arrived at this formula. – Gottfried Helms Dec 18 '17 at 21:30
  • @Kiyro6 - I added a link to my homepage for a small treatize on this style of proof shown (howeve not so simple) by John Simons - see http://go.helms-net.de/math/collatz/Collatz_1cycledisproof.pdf – Gottfried Helms Dec 18 '17 at 22:40