Abstract
In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth column of the infinite-row-six-column-matrix after a finite times operation, thus we convert “the 3X + 1 conjecture” into an equivalent conjecture, which is: Any positive integer n must become 1 after finite operations under formation of σ(n) , where Then, with the help of the infinite-row-four-column-matrix, we continue to use the recursive method to prove this conjecture strictly.
Highlights
The 3x + 1 conjecture is a world-famous unsolved number theory problem in recent 100 years [1]
By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth column of the infinite-row-six-column-matrix after a finite times operation, we convert “the 3X + 1 conjecture” into an equivalent conjecture, which is: Any positive integer n must become 1 after finite operations under formation of σ (n)
Lagarias wrote a review paper titled “The 3x + 1 problem and its generalizations”, saying: “in this paper I describe the history of the 3x + 1 problem and survey all the literature I am aware of about this problem and its generalizations” [2]
Summary
The 3x + 1 conjecture is a world-famous unsolved number theory problem in recent 100 years [1]. We give a summary proof: 1) If X ≠ 6(n −1) + 4 , according to proposition 2.5, any positive integer that is not 6(n −1) + 4 can be turned to 6(n −1) + 4 after a finite transformation according to F(x).
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