# The Collatz Conjecture : Unsolved but Useless The 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:

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. 