A Method of Finding All Existing Starting Numbers For Finite Arbitrarily Long Collatz Trajectories That Obey Any Behavior/Dynamics of Our Choosing.

We provide a general analytic formula to construct all existing starting odd numbers that obey our desired finite arbitrarily long Collatz trajectory, meaning that these starting odd numbers obey our pre-designated maximum factors of 2 at each iteration of the reduced Collatz map. We also provide another general analytic formula for finding the resulting odd numbers after N iterations of the reduced Collatz map. These formulas shed light on the structure of Collatz trajectories and other properties. We can also use this information to find in finite steps all existing Collatz trajectories that become 1 after any finite N iterations.We also will see that the ”location” of all of the 1’s in Collatz Conjecture can be found by solving a special case of the discrete log problem.