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This type of reasoning only gives you intuition about half the conjecture. The conjecture says two things: 1. No sequence grows bigger and bigger forever; and
by niknoble 6y ago
This type of reasoning only gives you intuition about half the conjecture. The conjecture says two things:
1. No sequence grows bigger and bigger forever; and
2. No sequence gets caught in a loop (except for the trivial 4 -> 2 -> 1 -> 4 loop).
Your article addresses 1, but I don't think it says much about 2. Sure, sequences should probably trend downward over time, but why shouldn't a sequence that hops upward at the beginning get snagged on a value it's already hit as it falls back down, trapping it a nontrivial loop?
- placebo 6y agoBoth parts you mentioned are derived from the Wikipedia formulation which is that "no matter what value of n, the sequence will always reach 1", but the subject of possible non trivial loops is an interesting one to contemplate from the coin toss perspective. The equivalence I made is to a truly random coin toss where heads means the number will be divided by two, and tails means the number will be multiplied by 1.5 and then 0.5 added to the result. If the coin toss is truly random, it stands to reason there will be no loop (following the same intuition that entropy always increases). Any permanent loop will contradict the randomness of the coin toss. Whether or not the reasoning above gives valid intuition to the conjecture being true depends upon the degree to which the Collatz mathematical procedure can be treated as generating a truly random process, and this is the point where I presented the main flaw in the intuition.