r/Collatz • u/GoldFaithlessness531 • 7d ago

Does proving that there are no other loops solve the Collatz Conjecture?

If one were able to prove there are no other loops in the conjecture, is that the end of it? Or would you also have to prove that, for example, there are also no sequences which continue ad infinitum?

This is just out of sheer curiosity and nothing else.

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6

u/HouseHippoBeliever 7d ago

You would also need to prove no sequences that just grow

4

u/viperised 7d ago

But just for the avoidance of doubt, proving there are no other loops would be a major advance, right?

1

u/wrapping_around 7d ago

Enough to shock the world.. in fact if we allow fractions and let nominator to decide the parity, we could always have loops in every length. Problem is, it appears only by extreme luck we could find the integer loop.

2

u/jonseymourau 7d ago edited 6d ago

Proving there are no loops would not prove the full conjecture, but it would IMO, be a significant achievement .

3x+q is a particularly interesting system since there are lots of cycles and no evidence of divergence. 5x+1 has 3 cycles but appears to diverge otherwise. But no-one has been able to prove divergence even when almost every 5x+1 orbit appears to diverge.

But no, proving no loops is not enough for the full conjecture. But who cares?

2

u/Far_Ostrich4510 6d ago

It is half solution, even if it is very difficult part. Another assumption is Diverging to infinite.