Collatz conjecture states that beginning with a positive integer, if one repeatedly performs the following operations to form a sequence of integers, the sequence will eventually reach the integer one; the operations being that if the integer is even, divide it by 2, but if the integer is odd, multiply it by 3 and add one; and also, use the result of each step as the input for the next step One would note the patterns of the sequence terms as the Collatz process reaches the equivalent powers, 2^(2k) (k = 2, 3,...) and the sequence reaches the integer 1 by repeated division by 2. Two main cases are covered. In Case 1, the integer can be written as a power of 2 as 2^(k) (k=1,2,3,u2026), and in this case, the sequence would reach the integer one by repeated division by 2, i.e., 2^(k-1), 2^(k-2), 2^(k-3,),u2026,2^(k-k). In Case 2, the integer cannot be written as a power of 2, but the sequence terms reach the equivalent power, 2^(2k) (k = 2, 3,...) and by repeated division by 2, the sequence will reach the integer 1. In Case 2, when the sequence terms reach some particular integers such as 5, 21 and 85, the application of 3n + 1 to these integers will result in the powers, 2^(2k). One would call these integers, the 2k-power converters. There are infinitely many 2k-power converters as there are 2^(2k) powers. There are infinitely many paths for converting integers to 2^(2k) powers. Of these paths, the integer 5-path, is the nearest 2^(2k) converter path to the integer 1 on the 2^(2k)-route. Other integers can follow the integer 5-path to 16 as follows: Let n be an integer whose sequence terms would reach 16, and let n ± r = 5, where r is the net change in the sequence terms before the integer 5; and one uses the positive sign if n<5, but the negative sign if n > 5. One will call the following, the 5-path 2k-converter formula: 3(n ± r) + 1 = 16. By the substitution axiom, using this formula, the sequence of every positive integer that cannot be written as a power of 2, would reach the integer, 16, and continue to reach the integer 1. Therefore, the sequence of every positive integer would reach the integer 1.