Number Theory

    

Collatz Sequence Proof (1st Way)

Authors: Taha M. Muhammad

1- Let x ∈ Loop of Collatz Sequence = LS(n), n, x,y,z,t,u,v ∈ N+ i) I proved LS(n) = { }, {x}, {x,y}, {x,y,z,t}, {x,y,z,t,u,u2026,v }, and {x,y,z,t,u,u2026,v,u2026} is false. ii) I proved LS(n) = {x,y,z} = {4,2,1} = {4,2,1, 4,2,1} =u2026= {4,2,1, u2026, 4,2,1} is true ∀n ∈ N+ 2- I created special sketch to find algebra expressions equal to x. 3- I created special Table to find values of elements in the LS(n). 4- I created a pattern to find values of x ∈ LS(n). 5- Observation: let r, k ∈ N+, r = number of elements in the LS(n):i) if r = 3k ⇒ x ∈ N+ ⇒ LS(n) = {1,2,4} ∀n ∈ N+ ii) if r = 3k + h, such that h=1,or 2⇒ x ∉ N+ ⇒ LS(n) has no loop ∀n ∈ N+ ∴ LS(n) ={4,2,1},∀n ∈ N+.

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[v1] 2025-06-12 22:24:29

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