Would it be possible to prove the collatz conjecture indirectly by demonstrating rules that apply to 'Collatz-like' conjectures? (I call anything where you simply change the values in the 3n+1 part of the conjecture to other values, holding everything else the same a Collatz-like conjecture) For instance if you could demonstrate that A. all infinitely increasing sequences of a Collatz-like conjecture follow [insert rule]. B. all sequences that loop in a Collatz-like conjecture either contain '1' as part of the loop, or else [insert condition]. C. when set to 3n+1, A requires that there be no infinitely increasing sequences, and B requires that all loops contain '1'. Would this prove the conjecture true? (I cannot think of an A and B that are both true and lead to C, but if someone found it, would it prove the conjecture?) Conversely if you had A and B as above, but instead of C had D. When set to 3n+1, A requires that there be at least one infinitely increasing set or E. When set to 3n+1, B requires that there be at least one set that loops and does not contain '1' would you be able to disprove the conjecture?