The Collatz Conjecture : Unsolved but Useless 1The Collatz Conjecture is one of the Unsolved problems in mathematics, especially in Number Theory. The Collatz Conjecture is also termed as 3n+1 conjecture, Ulam Conjecture, Kakutani’s Problem, Thwaites Conjecture, Hasse’s Algorithm, Syracuse Problem.


Statement:

Start with any positive integer.
Halve it, if it is even.
Or
triple it and add 1, if it is odd.

If you keep repeating this procedure, you shall reach the number 1 at last.

Illustrations

» Starting with 1 — we get 1 in first step.
» Starting with 2 (even) — we get 1 in second step and in one operation $ 2 \to 1 $
» Starting with 3 (odd) — we get 1 in 8th step $ 3 \to 10 \to 5 \to 16 \to 8 \to 4 \to 2 \to 1 $

Similarly, you can check this conjecture for every positive integer; you should get 1 at last according to this conjecture.

Mathematical Illustration

Let $ \mathbf {n} $ be a positive integer. Then it either be even or odd.
A. If n is even: Divide $ \mathbf {n} $ by $ \mathbf {2} $ and get $ \mathbf {\frac {n}{2} } $ . Is it 1? — conjecture applies on that positive integer. Again if it is even — redo the same work. If it is odd, then— see next step!
B. If n is odd: Multiply $ \mathbf n $ by $ \mathbf 3 $ & then add $ \mathbf 1 $ to find $ \mathbf 3n+1 $ . Is it 1? — conjecture applies on that positive integer. Again if it is even — redo the same work you did in A. If it is odd, then— redo the work of B!

Problem in this Conjecture

This conjecture has been tried on various kind of numbers, and those numbers have satisfied the Collatz Conjecture. But the question is that —

Is this conjecture applicable to every positive integer?

Note

Mathematicians have found no good use of Collatz Conjecture in Mathematics, so it is considered as a useless conjecture. But overall, it is unsolved — and we can’t leave any unknown or unsolved problems & principles in Math.

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