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Is it possible to find such a $l$,which that the known counter-example is only $l$, for any $k$, $f^k(n)≠1$? [Is there a modified Collatz rule 3n+$l$ where there is only one loop or "counter-example" and is not 3n+1?] $$f(n) = \begin{cases} n/2 &\text{if } n \equiv 0 \pmod{2}\\ 3n+l & \text{if } n\equiv 1 \pmod{2} .\end{cases}$$

Here, $l$ is odd number and $l ≠1$

Because, for $n=l$ , $l$ is counterexample.

For example: $l=7$

The known counterexample are $(5,1),(7,7).$ [According to P..] [In other words, 3n+7 has two loops or two "counter-examples": 5-22-11-40-20-10-5... and 7-28-17-7...]

But, I'm looking for the known counterexample for function $f(n)$ is only $l$. I have tried many. But I did not succeed. Is there such a $l$ ?

If the question is unclear, please edit... I did everything I could.

Griffon Theorist697
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Soru
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    What is it a counter-example to? This question is missing some context? Can you supply a more clear statement of what you are looking for? This looks like it is related to the Collatz conjecture, but it is unclear. – Thomas Andrews Oct 20 '17 at 14:21
  • @Thomas Andrews I'm trying to edit.... – Soru Oct 20 '17 at 14:24
  • Are you trying to find loops aside from the 4-2-1 loop for 3n+1 in different rules such as 3n+5? If so, I may be able to answer your question. – Griffon Theorist697 Oct 20 '17 at 14:30
  • @Thomas Andrews I did everything I could... – Soru Oct 20 '17 at 14:31
  • OP, if you speak English as a second language, maybe you could ask someone to interpret your question. As it stands, it's really not clear what you're asking :) – Jam Oct 20 '17 at 14:33
  • @Griffon Theorust697 I'm looking for counterexample is only $l$. – Soru Oct 20 '17 at 14:33
  • @Jam yes, I use google translate. – Soru Oct 20 '17 at 14:34
  • Hmmm... for 3n+5, n=5, 3(5)+5 = 20, 20/2 = 10, 10/2 = 5... The loop is 20-10-5 and 5=n. Is 20-10-5 a counterexample? – Griffon Theorist697 Oct 20 '17 at 14:40
  • I sorry, but what does "looking for counterexample is only $l$" mean? Counterexample to what? – Thomas Andrews Oct 20 '17 at 14:40
  • We can't edit your question because we'd need to know what your question means to make it more clear. – Thomas Andrews Oct 20 '17 at 14:41
  • @ThomasAndrews I don't think "Counterexample" here means a literal counterexample to the Collatz Conjecture. I think it means he is trying to find a number that doesn't work with his equation, which is a little tricky because I believe he is working with modified Collatz rules such as 3x+5 and 3x+7. I know in 3x+7 there is a 5-22-11-40-20-10-5-... loop. – Griffon Theorist697 Oct 20 '17 at 14:46
  • @Thomas Andrews Counterexample must be only $l$. – Soru Oct 20 '17 at 14:48
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    Sigh, I know it must be only $l$. What statement is $l$ supposed to be a counter-example to? – Thomas Andrews Oct 20 '17 at 14:49
  • İt must be a "Counterexample" for $f(n)$ – Soru Oct 20 '17 at 15:01

1 Answers1

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There are two rules I am aware of that provide 1 "counterexample" for $l$:

$3n+3$ and $3n+9$

It is unproven if these are the only "counterexamples" for all numbers because we do not know if the Collatz Conjecture is true. However,

I checked both $3n+3$ and $3n+9$ for 1,000,000 starting numbers.

There was only 1 "counterexample" or loop for each:

$3n+3$: 12-6-3

$3n+9$: 36-18-9

Note: The files I uploaded are huge. You may need to download them.

I used some code I already wrote in Java that keeps track of the number of cycles and then dump the results into a .csv file. Since Excel can only hold so may numbers, I restricted my sample size to 1,000,000.

Gottfried Helms also worked on this problem and goes into greater detail here.

Griffon Theorist697
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