M-cycles Cannot Occur
The conjecture could be proven, indirectly, as a consequence of the following:
- no infinite divergent trajectory occurs
- no cycle occurs (apart from the trivial (1,2) cycle)
These being true, all natural numbers would have a trajectory down to one.
In 1977, R. Steiner, and in 2000 and 2002, J. Simons and B. de Weger (based on Steiner's work), proved the nonexistence of certain types of cycles.
Read more about this topic: Collatz Conjecture
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“Usually he has no thoughts—but as an exception to this rule sometimes nasty thoughts occur to him.”
—Friedrich Nietzsche (1844–1900)