Theoretical Mathematics & Applications

Analysis of Collatz Conjecture Rules

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• Abstract

   

  A proof of the Collatz Conjecture is presented. Changing the perspective of the problem from looking at the pattern of the positive integers to looking at the conjecture rules made the proof possible. The conjecture rule for even numbers organizes all positive integers into unique sets and the rule for odd numbers interconnects the unique sets into dendritic pathways to “1.” Infinite loops, other than the minor 4-2-1 loop after reaching “1,” and values continually increasing to infinity are shown to be mathematically impossible. The proof predicted a general equation that shows all positive integers reach a final value of “1," and calculates the values and locations of the odd positive integers during the iterations for each tested positive integer.

   

  JEL classification numbers: CO2; C65.

  Keywords: Collatz Conjecture, Proof, Rules, Dendritic pathway.