A very different property of numbers: Can be changed to $1$ by applying only these two operations.

While playing with numbers, I thought of type of numbers, and then the first thing came into mind was $\text{Odd}$ and $\text{Even}$.

I observed a very interesting fact that any $x\in\Bbb{N}$ can be converted into $1$ by applying these two operations.

$\text{If the number is odd:}$ $$\text{Multiply it by 3 and add 1}\tag{3k+1}$$

$\text{If the number is even:}$ $$\text{Divide it by 2}\tag{\frac{k}{2}}$$

For Example: $$6\tag{Even}$$ $$\frac62=3\tag{Odd}$$ $$3\times3+1=10\tag{Even}$$ $$\frac{10}2=5\tag{Odd}$$ $$3\times5+1=16\tag{Even}$$ $$\frac{16}2=8\tag{Even}$$ $$\frac{8}2=4\tag{Even}$$ $$\frac{4}2=2\tag{Even}$$ $$\frac{2}2=1$$ I checked it $2-30$ and found my observation true. Then by taking a general case by letting the number $k$ but can't reach on any result since it form a infinite series of operations.

Please help me prove this or help me disprove it by giving a counter example!!!

• – Andreas Blass Apr 25 '17 at 19:01
• thanks for the link @AndreasBlass – Harsh Kumar Apr 26 '17 at 2:26