1

Context: Collatz conjecture

What I call a 'branch number', is a number accessible by 2 different routes.

Example :

  • 24 is not a branch number, it can be accessed only from 48 (division by 2)
  • 16 is a branch number, it can be accessed from 32 (divison by 2) or 5 (3x+1)

Is it possible to find a formula that generates these numbers or is this tied to the problem itself - so solving this would resolve the problem?

Thanks

Update

I'm talking about finding a function that generates these numbers with this sequence :

[10, 16, 22, 28, 34, 40, 46, ...]

toto
  • 201
  • 7
    Unsolicited advice: Stay away from the horrific time-sink which is this conjecture. – Matt Aug 21 '18 at 13:22
  • 1
    What @Matt said above. This isn't something that can be resolved with simple algebraic manipulation. – anomaly Aug 21 '18 at 13:27
  • How would it help resolve the problem? Can you elaborate a little more in your question? Also what @Matt said, Paul Erdos (a famous mathematician of the 20th century) once said "Mathematics is not yet ready for such problems", and in my very humble opinion I don't think much has changed since he said that. – Ltoll Aug 21 '18 at 13:32
  • 1
    Your numbers are simply $6k+4$ for $k=1,2,3,4...$. – Jaap Scherphuis Aug 22 '18 at 09:10
  • @JaapScherphuis thanks – toto Aug 22 '18 at 09:17

2 Answers2

0

We know that odd numbers cannot be branch numbers, because they can only be generated in one way (i.e., dividing by 2). Even numbers can only be branch numbers if they are of the form $3x+1$ for some $x$. This means that all positive even numbers $e$ that satisfy $e\equiv1\mod{3}$ are branch numbers.

Rushabh Mehta
  • 13,663
0

[10,$\qquad$ 16,$\quad$ 22,$\qquad$ 28,$\quad$ 34,$\qquad$ 40,$\qquad$ 46, ...] can be rewritten as
[1*6+4, 2*6+4, 3*6+4, 4*6+4, 5*6+4, ...] . You surely see the pattern - and the general formula?

Gottfried Helms
  • 34,920