Journal Entries

The 3n+1 Conjecture

Today I developed a formula for the probability that the sequence starting with an odd natural number divisible by 3 is bounded (does not get arbitrarily large). There's a very small probability that the sequence is unbounded. Since the formula models the data so well, a random process appears to determine whether the sequence is bounded and it's likely that this part of the 3n+1 conjecture is unprovable.

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Latest reply: Dec 27, 2008

The 3n+1 Problem

I made significant progress on proving part of the 3n+1 conjecture today (the part that says no sub-sequence other than {4, 2, 1} repeats). A major theorem (not proved yet) is; The limbs ending in an element of S1 are either dead, one-element limbs, two-element limbs, or four-element limbs. (See the Guide Entry "An Approach to the 3n+1 Problem" for the definition of these terms.) An elaborate induction argument would be required to prove this. I'm just now proving some rudimentary theorems about least-residue trees. Much work remains to be done, but it's clear that there's nothing profound or unprovable about this part of the 3n+1 conjecture.

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Latest reply: Dec 18, 2008


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smallfrey

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