Number Theory


On the Impossibility of the Existence of Cycles Including Integer Multiples of 3 in a Collatz Sequence

Authors: Ho Soo Shin

As is well known, the Collatz sequence, which is also named as the hailstone sequence, follows the rule of Collatz conjecture. The rule requires us to divide any positive even integer by 2. We must multiply every positive odd number by 3 and then add 1 according to the rule. By investigating residues modulo 3, I will prove any integer multiple of 3 cannot appear more than one time in a Collatz sequence, which implies any multiple of 3 cannot be included in a possible cycle of the Collatz sequence.

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[v1] 2018-12-20 21:07:15

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