**3n+1 conjecture, Ulam Conjecture, Kakutani’s Problem, Thwaites Conjecture, Hasse’s Algorithm, Syracuse Problem.**

### Statement:

Start with any positive integer.

•Halveit, if it is even.

Or

•tripleit andadd 1, if it is odd.If you keep repeating this procedure, you

shallreach 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.