Collatz Sequence Proof (1st Way)

The Collatz sequence is defined for any positive integer 𝑛 by repeatedly applying two rules: if 𝑛 is even, compute 𝑛 / 2 ; if 𝑛 is odd, compute 3 𝑛 + 1 . Let the sequence be 𝑆 ( 𝑛 ) = { 𝑎 , 𝑏 , 𝑐 , … , 𝑥 , … , 𝑡 } . This work examines whether any loop other than the known { 4 , 2 , 1 } cycle can occur. Assume a positive integer 𝑥 lies in a hypothetical loop of length 𝑟 ∈ 𝑁 + . Applying the Collatz rules to 𝑥 and solving the resulting loop equations shows that the only possible values of 𝑥 are 1 , 2 , 4 , which correspond exactly to the classical loop { 4 , 2 , 1 } . A general algebraic form for loop elements further demonstrates that when 𝑟 ≡ 0 ( m o d 3 ) , the only solution is 𝑥 = 1 , again yielding the same loop. Since any Collatz sequence satisfies 𝑙 𝑆 ( 𝑛 ) = 𝑙 𝑆 ( 𝑥 ) , and the only valid loop element is 𝑥 = 1 , it follows that every Collatz sequence contains exactly one loop: { 4 , 2 , 1 } .