The Collatz Conjecture is an unsolved math problem first thought of by Lorath Collatz. The basic idea behind it is that all numbers will boil down to 1 under the conjecture. The collatz conjecture tackles the problem about how the prime factorization of "a" affect the prime factorization of "a+1"? So it gives us more insight into primes. Collatz sequences are also known as hailstone sequences, as they can bounce up and down much like hailstones in a cloud were thought to. Despite hundreds of published papers on the Collatz conjecture, and many more unpublished works (including countless failed proofs), the conjecture remains unsolved today.
Number of steps:
Collatz String:
$$a_n = \begin{cases} \frac{1}{2} a \scriptscriptstyle n-1 &\text{if } a \scriptscriptstyle n-1\displaystyle \mod 2 =0 \\ 3a \scriptscriptstyle n-1 \displaystyle +1&\text{if } a \scriptscriptstyle n-1\displaystyle \mod 2 =1 \\ \end{cases} $$
A key difficulty with the Collatz iteration is that it can greatly distort the distribution of a set of numbers – some numbers collide into each other, others get skipped entirely. Because of this, it's really hard to deal with the data points for each number.