Collatz Conjecture

Take a number number $ n $ (non-zero positive integer), if $ n $ is even, divide it by $ 2 $, else multiply by $ 3 $ and add $ 1 $. Start over with the result until you get $ 1 $.

Mathematically the algorithm is defined by the function $ f $: $$ f_{3n+1}(n) = \begin{cases}{ \frac{n}{2}} & {\text{if }}n \equiv 0 \mod{2} \\ 3n+1 & {\text{if }} n \equiv 1 \mod{2} \end{cases} $$

Example: $ n=10 $, $ 10 $ is even, divide it by $ 2 $ and get $ 5 $,
$ 5 $ is odd, multiply it by $ 3 $ and add $ 1 $ to get $ 16 $,
Continue the sequence to get $ 8 $, $ 4 $, $ 2 $ and $ 1 $.

The Collatz conjecture stipulates that the 3n+1 algorithm will always reach the number 1.

Some numbers have surprising trajectories like 27, 255, 447, 639 or 703

No, nobody has found a number for which it does not work but nobody has found any mathematical proof that the conjecture is always true.

This is why the conjecture is also called the Syracuse problem or the Collatz problem and it is not a theorem.

A number never appears twice in the sequence.

Any sequence ends with a series of powers of 2.

An odd number is always followed by an even number.

The numbers 5 and 32 give the same result.

// Javascript Algorithm
function step(n) {
if (n%2 == 0) return n/2;
return 3*n+1;
}
function collatz(n) {
var nb = 1;
while (n != 1) {
n = step(n);
nb++;
}
return nb;
}
// Python Code Algo
def collatz(x):
while x != 1:
if x % 2 > 0:
x =((3 * x) + 1)
list_.append(x)
else:
x = (x / 2)
list_.append(x)
return list_


The Collatz conjecture is also known as

- 3n + 1 conjecture

- Ulam conjecture

- Kakutani's problem

- Thwaites conjecture

- Hasse's algorithm

- Syracuse problem

This table is for numbers until 1000 (total time => numbers)

This table shows numbers until 1000 (max number reached => numbers)

Formulated in 1937 it remains unsolved: nobody has been able to prove this conjecture always ends with 1.