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You would first have to prove that the number is actually a counterexample. I don't know what you mean by calculating the whole sequence, as it would be an infi
by cyborgx7 7y ago
You would first have to prove that the number is actually a counterexample. I don't know what you mean by calculating the whole sequence, as it would be an infinite sequence. Proving a counterexample seems like a halting problem kind of problem.
- bluGill 7y agoErr... right. I stand corrected.
- cyborgx7 7y agoBut the question of a counterexample (eventhough it probably doesn't exist) is interesting. Is there a number for which we don't know weather it is a counterexample because it would take too long to compute? I feel like any single counterexample you'd propose would be proven wrong pretty quickly.
- bluGill 7y agoThe problem is if my counter example is a counter example it diverges to infinity. (someone else proposed a proof that there are no cycles other than at 1, if this proof is wrong my counter example could be a cycle and thus "easy" to show). However if goes to infinity it is really hard to see how you show it doesn't eventually converge if you just went a little big longer. Note that every counter example you propose is actually a sequence of counter examples. It would be interesting to examine the properties of whatever numbers that counter example has in common. Though this is an obvious thing that I suspect someone has already done to no effect.
- ifdefdebug 7y agoUnless you find a counterexample which results in a loop N1 -> N2 -> ... -> Nn -> N1