Proving Collatz Conjecture by finding cycles and diverging seeds

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Abstract The Collatz conjecture can be formulated in this question: Do we guarantee reaching one after applying function f (Hereinafter called Collatz function) multiple times? f(x) = (3x+1, x|2) ∨ (x|2, x ∤ 2) In this paper, we discuss an approach to analyze Collatz conjecture nature for all qx + 1, q ∈ {1,3,5,7,..}. this approach builds a method to know the number of potential cycles for q x + 1, q ∈ {1,3,5,7,..}, x < Mx, Mx ∈ ℂ. Multiple non-trivial cycles (numbers that form a cycle and does not reach one) were previously found where q ∈ {5, 181}. This paper will introduce equations that find non-trivial cycles, and therefore proving or disproving Collatz conjecture for any q.

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